diff --git "a/vocab.txt" "b/vocab.txt"
new file mode 100644--- /dev/null
+++ "b/vocab.txt"
@@ -0,0 +1,65004 @@
+<unk>
+<bos>
+<eos>
+<pad>
+<think_token>
+</think_token>
+<thought>
+</thought>
+<memory>
+</memory>
+<intent>
+</intent>
+<reflection>
+</reflection>
+<metacognition>
+</metacognition>
+<reasoning>
+</reasoning>
+<analysis>
+</analysis>
+<start:math>
+<end:math>
+<math>
+</math>
+<proof>
+</proof>
+<assume>
+</assume>
+<deduce>
+</deduce>
+<lemma>
+</lemma>
+<theorem>
+</theorem>
+<corollary>
+</corollary>
+<definition>
+</definition>
+<axiom>
+</axiom>
+<qed>
+<logic>
+</logic>
+<forall>
+<exists>
+<not>
+<and>
+<or>
+<implies>
+<iff>
+<contradiction>
+<tautology>
+<fallacy>
+<premise>
+</premise>
+<conclusion>
+</conclusion>
+<inference>
+</inference>
+A
+B
+C
+D
+E
+F
+G
+H
+I
+J
+K
+L
+M
+N
+O
+P
+Q
+R
+S
+T
+U
+V
+W
+X
+Y
+Z
+a
+b
+c
+d
+e
+f
+g
+h
+i
+j
+k
+l
+m
+n
+o
+p
+q
+r
+s
+t
+u
+v
+w
+x
+y
+z
+ 
+.
+,
+!
+?
+-
+_
+:
+;
+'
+"
+/
+\
+@
+#
+$
+%
+^
+&
+*
+(
+)
+[
+]
+{
+}
+=
++
+|
+~
+`
+<sp1>
+<sp2>
+<sp3>
+<sp4>
+<sp5>
+<sp6>
+<sp7>
+<sp8>
+the
+and
+or
+but
+in
+on
+at
+to
+for
+of
+with
+by
+you
+he
+she
+it
+we
+they
+hello
+world
+test
+example
+. 
+2
+1
+frac
+is
+3
+**
+4
+0
+5
+that
+sqrt
+The
+6
+are
+number
+be
+boxed
+text
+right
+left
+cdot
+times
+8
+can
+7
+10
+as
+<
+9
+Therefore
+find
+from
+equation
+sin
+have
+pi
+triangle
+angle
+each
+>
+Since
+which
+this
+cos
+total
+Let
+12
+two
+This
+Given
+given
+value
+point
+We
+circ
+an
+sum
+Thus
+all
+one
+alpha
+If
+then
+numbers
+For
+To
+function
+line
+overrightarrow
+a_
+first
+15
+quad
+20
+has
+not
+leq
+2x
+both
+points
+where
+answer
+cn_k12
+these
+there
+get
+16
+need
+area
+must
+problem
+Now
+100
+such
+into
+square
+than
+25
+end
+sides
+if
+circle
+30
+implies
+11
+so
+Conclusion
+values
+begin
+geq
+length
+AB
+set
+terms
+sequence
+side
+So
+x_
+between
+will
+any
+positive
+equal
+per
+when
+60
+correct
+solve
+integer
+three
+many
+13
+distance
+its
+dfrac
+50
+let
+theta
+textbf
+formula
+synthetic_math
+18
+second
+In
+Total
+Hence
+using
+Find
+ldots
+calculate
+since
+possible
+expression
+inequality
+same
+ABC
+gives
+24
+40
+orca_math
+Calculate
+olympiads
+real
+3x
+form
+cases
+14
+digit
+term
+mathbf
+BC
+final
+a_n
+equiv
+other
+beta
+From
+condition
+a_1
+time
+integers
+cost
+What
+their
+2n
+First
+only
+solution
+probability
+When
+common
+up
+mathbb
+students
+infty
+out
+through
+x_1
+90
+Using
+2a
+36
+sum_
+least
+product
+range
+hours
+prime
+odd
+qquad
+ratio
+AC
+parallel
+Solving
+how
+maximum
+tan
+By
+also
+Step
+45
+17
+leqslant
+speed
+price
+more
+even
+amount
+pmod
+original
+overline
+know
+digits
+align
+after
+equations
+cm
+does
+x_2
+cdots
+use
+ln
+minimum
+determine
+because
+pmatrix
+following
+angles
+S_
+How
+follows
+radius
+denote
+roots
+4x
+units
+base
+minutes
+step
+lines
+divisible
+quadratic
+some
+means
+remaining
+was
+respectively
+21
+average
+difference
+day
+would
+different
+four
+new
+binom
+pm
+km
+Then
+180
+triangles
+neq
+us
+Substituting
+Determine
+meters
+axis
+less
+days
+conditions
+plane
+his
+no
+120
+center
+interval
+27
+smallest
+properties
+ways
+perpendicular
+Each
+Consider
+consider
+ab
+There
+32
+negative
+a_2
+exactly
+non
+tangent
+squares
+75
+hour
+question
+Simplifying
+80
+Rs
+vertices
+mathrm
+thus
+19
+height
+coordinates
+Rightarrow
+true
+geqslant
+log_
+2k
+holds
+zero
+option
+Area
+rate
+35
+simplifies
+arithmetic
+Simplify
+According
+third
+intersection
+divide
+years
+fraction
+increasing
+segment
+graph
+vertex
+000
+pairs
+them
+her
+2y
+64
+miles
+root
+part
+every
+geometric
+AD
+people
+train
+distinct
+Solution
+A_
+greater
+200
+It
+satisfy
+Number
+synthetic_amc
+ax
+volume
+system
+22
+28
+half
+certain
+feet
+rectangle
+start
+1000
+Prove
+CD
+multiple
+Substitute
+red
+Case
+statement
+midpoint
+what
+within
+simplify
+case
+were
+Next
+satisfies
+work
+48
+always
+should
+omega
+denominator
+calculated
+divided
+b_
+150
+2b
+slope
+π
+y_
+cannot
+lambda
+solutions
+solving
+weight
+result
+ellipse
+However
+inches
+being
+percentage
+add
+polynomial
+last
+array
+approx
+exists
+C_
+largest
+23
+pair
+cube
+opposite
+either
+--
+relationship
+therefore
+5x
+who
+70
+factor
+complex
+valid
+year
+tests
+lengths
+back
+After
+parabola
+coordinate
+initial
+functions
+lies
+intersect
+lfloor
+remainder
+constant
+our
+26
+rfloor
+Delta
+had
+above
+circles
+becomes
+steps
+xy
+take
+elements
+balls
+age
+prove
+basic
+smaller
+choose
+substitute
+together
+×
+intersects
+row
+water
+unit
+theorem
+based
+49
+Analysis
+BD
+mid
+factors
+72
+do
+decreasing
+formed
+takes
+log
+ABCD
+seconds
+write
+quadrilateral
+gamma
+Define
+aligned
+33
+Solve
+power
+incorrect
+x_0
+subtract
+kg
+vec
+300
+Identify
+00
+segments
+show
+results
+2m
+500
+symmetry
+proposition
+sets
+much
+property
+vectors
+books
+without
+varphi
+cap
+choice
+cup
+twice
+perimeter
+Part
+a_i
+person
+group
+over
+3a
+Sum
+6x
+general
+blue
+make
+derivative
+perp
+information
+a_3
+42
+draw
+natural
+profit
+did
+81
+multiply
+le
+b_n
+among
+obtain
+29
+31
+passes
+Assume
+hyperbola
+another
+week
+order
+_1
+necessary
+defined
+equals
+box
+inside
+before
+curve
+AM
+360
+4a
+now
+express
+down
+y_1
+parts
+perfect
+corresponding
+_n
+known
+hence
+making
+Time
+bc
+coefficient
+five
+Similarly
+S_n
+sold
+yuan
+needs
+analyze
+3y
+used
+lie
+discount
+isosceles
+white
+sub
+cards
+dollars
+Distance
+400
+coefficients
+div
+Option
+place
+sufficient
+calculation
+about
+consecutive
+contains
+Combining
+decimal
+...
+money
+black
+Speed
+considering
+inequalities
+Finally
+99
+next
+definition
+mean
+like
+monotonically
+dx
+Suppose
+expressed
+vector
+approximately
+larger
+found
+Calculating
+spent
+equilateral
+He
+edges
+position
+diagonal
+_2
+rightarrow
+pieces
+diameter
+color
+2i
+56
+diagonals
+55
+3n
+PQ
+regular
+student
+complete
+method
+long
+required
+mod
+most
+Here
+Equation
+similar
+’
+standard
+origin
+due
+label
+additional
+increase
+class
+further
+adjacent
+34
+written
+α
+A_1
+along
+37
+pounds
+aops_forum
+hand
+65
+b_1
+sphere
+th
+represents
+4y
+width
+sums
+series
+period
+On
+fourth
+interest
+domain
+combined
+assume
+54
+symmetric
+gcd
+numerator
+faces
+edge
+identity
+modulo
+conclude
+areas
+At
+y_2
+Adding
+under
+score
+examines
+divisor
+exist
+passing
+choices
+could
+type
+coins
+check
+Theorem
+Express
+drawn
+small
+large
+Finding
+blacksquare
+dots
+share
+apples
+while
+expressions
+inscribed
+face
+focus
+bisector
+divisors
+marbles
+greatest
+ensure
+4k
+leads
+125
+Problem
+hr
+distances
+dividing
+taken
+games
+144
+quadrant
+boys
+x_i
+selling
+made
+bx
+combinations
+convert
+Set
+directly
+divides
+boxes
+trigonometric
+surface
+operatorname
+around
+bought
+car
+starting
+hold
+600
+denoted
+yields
+specific
+minute
+cells
+month
+school
+ball
+meet
+Point
+respect
+polygon
+pattern
+subset
+As
+understanding
+44
+8x
+positions
+percent
+six
+lg
+considered
+options
+already
+Two
+rewrite
+degree
+green
+Calculation
+entire
+Points
+2c
+sine
+false
+F_
+single
+Use
+Compute
+52
+median
+chosen
+whose
+contradiction
+leading
+ge
+application
+She
+conclusion
+whole
+sequences
+binomial
+grid
+player
+fractions
+41
+multiplying
+represent
+outcomes
+resulting
+cosine
+Pythagorean
+finding
+simplifying
+a_4
+substituting
+unique
+x_n
+column
+85
+rectangular
+apply
+DE
+expansion
+dot
+shown
+your
+Combine
+63
+needed
+covered
+2023
+still
+increases
+simplified
+team
+fixed
+absolute
+rho
+change
+count
+game
+240
+Applying
+96
+phi
+39
+table
+cross
+cubes
+equality
+round
+involves
+costs
+obtained
+initially
+Setting
+C_1
+top
+selected
+2t
+size
+rule
+cycle
+Expanding
+Final
+girls
+ac
+3b
+parallelogram
+states
+key
+move
+represented
+follow
+Triangle
+2000
+way
+powers
+P_
+adding
+man
+liters
+Subtract
+division
+rational
+old
+Length
+imaginary
+3k
+EF
+element
+tax
+forms
+degrees
+intervals
+4n
+region
+linear
+int_
+°
+operation
+multiples
+makes
+marks
+These
+moves
+a_k
+corresponds
+84
+arc
+B_
+satisfying
+equivalent
+Cost
+cone
+see
+circumcircle
+lt
+x_3
+provided
+again
+MN
+direction
+compute
+want
+sales
+chord
+II
+those
+full
+operations
+paid
+250
+cubic
+hypotenuse
+AE
+once
+including
+b_2
+tickets
+groups
+discriminant
+pages
+OA
+Initial
+plus
+pass
+moving
+38
+tank
+51
+contain
+43
+players
+children
+cover
+T_
+figure
+2016
+67
+rows
+①
+Understanding
+Considering
+AP
+min
+trapezoid
+160
+Which
+All
+months
+One
+statements
+added
+gallons
+Probability
+xi
+items
+trip
+47
+Notice
+event
+Percentage
+sign
+②
+buy
+matches
+Note
+shows
+process
+John
+midpoints
+placed
+tetrahedron
+occurs
+800
+acute
+With
+LCM
+kmph
+data
+match
+Divide
+Angle
+colors
+vertical
+correctly
+received
+cents
+A_2
+101
+card
+Evaluate
+convex
+runs
+previous
+polar
+break
+remains
+cookies
+95
+kx
+a_5
+double
+planes
+random
+CA
+cot
+subsets
+determined
+2ab
+containing
+105
+members
+monotonicity
+Check
+OB
+B_1
+Apply
+9x
+225
+variable
+current
+______
+primes
+empty
+pyramid
+7x
+collinear
+products
+loss
+2017
+3600
+during
+PA
+4b
+110
+deduce
+hexagon
+formulas
+factorization
+gt
+BE
+distribution
+constraints
+Multiply
+interior
+highest
+Analyze
+income
+cyclic
+indeed
+well
+cut
+log_2
+140
+10x
+46
+friends
+matrix
+gets
+subtracting
+away
+involving
+bag
+measure
+salary
+ounces
+bar
+taking
+congruent
+meaning
+starts
+level
+nearest
+store
+cars
+path
+abc
+2r
+evaluate
+middle
+br
+98
+having
+reach
+kilometers
+210
+66
+a_0
+yellow
+PB
+applying
+relation
+men
+256
+intersecting
+3m
+board
+transformation
+includes
+cell
+relative
+max
+BAC
+answers
+scores
+analysis
+piece
+900
+centers
+Also
+zeros
+horizontal
+votes
+Condition
+trees
+identify
+≈
+dimensions
+Subtracting
+Summing
+k_
+rewritten
+fill
+Start
+variables
+mathcal
+arrangements
+Multiplying
+Proof
+induction
+randomly
+math
+foci
+altitude
+legs
+configuration
+below
+book
+select
+z_
+p_
+53
+been
+2d
+Second
+pay
+combination
+proceed
+letters
+exponents
+2p
+requires
+itself
+widehat
+eccentricity
+propositions
+ll
+y_0
+Consequently
+Among
+GM
+Dividing
+multiplication
+circumference
+108
+giving
+r_1
+V_
+across
+scored
+ensuring
+Both
+centered
+population
+derived
+58
+bottom
+hline
+S_1
+events
+pencils
+fact
+Show
+integral
+CP
+cylinder
+An
+problems
+calculations
+arrange
+off
+increased
+projection
+Inequality
+list
+scenario
+Law
+4m
+centimeters
+They
+DC
+2015
+mx
+135
+57
+space
+Analyzing
+fish
+reasoning
+New
+observe
+consists
+Statement
+foot
+places
+travel
+took
+straight
+derive
+AF
+according
+relationships
+angled
+p_1
+inverse
+call
+3i
+magnitude
+note
+128
+78
+assumption
+read
+include
+Average
+give
+l_
+section
+city
+No
+ends
+nor
+questions
+mass
+Line
+done
+③
+negation
+say
+outside
+1200
+drawing
+identical
+gave
+b_i
+12x
+Convert
+Additionally
+wins
+coin
+r_2
+62
+traveled
+house
+c_
+bus
+oranges
+Setup
+′
+Based
+mu
+rest
+simple
+88
+T_n
+C_2
+earnings
+Cartesian
+whether
+until
+450
+sample
+rectangles
+grams
+isolate
+θ
+room
+shaded
+implying
+lim_
+identities
+play
+similarity
+calculating
+define
+appears
+onto
+Circle
+candies
+rules
+Bob
+permutations
+t_
+setting
+Monday
+principle
+mph
+ones
+changes
+strictly
+÷
+eliminate
+neither
+S_2
+radii
+mainly
+tangents
+68
+cups
+f_
+progression
+medium
+minus
+verify
+fifth
+indicating
+individual
+shape
+Base
+Profit
+z_1
+clear
+him
+Taking
+got
+720
+ensures
+addition
+2018
+exponent
+necessarily
+prod_
+mixture
+61
+good
+eggs
+05
+arrangement
+weeks
+reaches
+4ac
+ABD
+connected
+candy
+2010
+OC
+die
+999
+a_6
+marked
+employees
+heads
+Price
+leaves
+Construct
+asy
+5y
+ell
+sector
+OP
+sim
+2z
+contradicts
+ordered
+intersections
+corner
+Volume
+BM
+Alice
+3000
+∈
+decrease
+colored
+arranged
+91
+PM
+2R
+Possible
+625
+59
+130
+tangency
+de
+Counting
+…
+Rearranging
+paper
+computational
+limit
+sigma
+Plugging
+Rearrange
+desired
+normal
+probabilities
+76
+assuming
+rhombus
+2020
+meter
+ten
+tree
+Add
+winning
+monthly
+columns
+investment
+CE
+floor
+2022
+tens
+Mean
+wants
+216
+multiplied
+Count
+combining
+prism
+revenue
+sup
+compare
+And
+central
+blocks
+letter
+1500
+Starting
+shirts
+friend
+conjugate
+PC
+exact
+field
+77
+internal
+independent
+representation
+approach
+indicates
+sale
+yz
+Choose
+p_2
+pentagon
+centroid
+aim
+expected
+impossible
+vmatrix
+XY
+dice
+cans
+2021
+teams
+win
+partial
+Assuming
+looking
+food
+2019
+paint
+tiles
+simplification
+97
+Ratio
+forming
+longer
+triangular
+female
+BCD
+located
+annual
+excluding
+company
+O_1
+reciprocal
+rotation
+bags
+Answer
+similarly
+actual
+daily
+divisibility
+Because
+d_
+≤
+2014
+complement
+bisectors
+combine
+Factor
+covers
+bases
+alone
+quotient
+home
+higher
+bottles
+subseteq
+2A
+89
+simultaneously
+meets
+consistent
+connecting
+proof
+Intersection
+2013
+leg
+ratios
+platform
+boldsymbol
+lower
+73
+just
+Rewrite
+parameter
+odot
+except
+bridge
+purchase
+4i
+satisfied
+semi
+2xy
+Vieta
+Verify
+uses
+proportion
+go
+present
+β
+capacity
+regions
+women
+pens
+milk
+mn
+2f
+Square
+Triangles
+fewer
+algebraic
+，
+filled
+works
+proportional
+④
+intercept
+ca
+polynomials
+2011
+lateral
+choosing
+job
+numbered
+mathematical
+infinite
+construct
+5a
+circular
+Expand
+z_2
+decreases
+BP
+Properties
+travels
+run
+Verification
+ages
+92
+equally
+Expression
+Checking
+axes
+understand
+ACB
+6y
+2ax
+weights
+ability
+O_
+r_
+CB
+Three
+dogs
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