We are given three congruence equations involving integers $a$, $b$, and $c$ and the modulus 7.
The first equation is $abc\equiv 1\pmod 7$, which means that the product of $a$, $b$, and $c$ is congruent to 1 modulo 7.
The second equation is $5c\equiv 2\pmod 7$, which means that $5c$ is congruent to 2 modulo 7.
The third equation is $6b\equiv 3+b\pmod 7$, which means that $6b$ is congruent to $3+b$ modulo 7.

To find the remainder when $a+b+c$ is divided by 7, we need to determine the value of $X$.

Let's analyze the second equation, $5c\equiv 2\pmod 7$.
Since the modulus is 7, we can find the possible values of $c$ by multiplying the integers from the set of positive integers less than $X$ by 5 and finding the remainder when divided by 7.
Looking at the given information, we know that the remainder when $5c$ is divided by 7 is 2.
From the set of positive integers less than $X$, we can try different values of $c$ and check if $5c$ gives a remainder of 2 when divided by 7.

Let's try $c=1$:
$5(1) = 5$, which gives a remainder of 5 when divided by 7.

Let's try $c=2$:
$5(2) = 10$, which gives a remainder of 3 when divided by 7.

Let's try $c=3$:
$5(3) = 15$, which gives a remainder of 1 when divided by 7.

Let's try $c=4$:
$5(4) = 20$, which gives a remainder of 6 when divided by 7.

Let's try $c=5$:
$5(5) = 25$, which gives a remainder of 4 when divided by 7.

Let's try $c=6$:
$5(6) = 30$, which gives a remainder of 2 when divided by 7.

From the above calculations, we can see that $c=6$ satisfies the second equation $5c\equiv 2\pmod 7$.

Now, let's analyze the third equation, $6b\equiv 3+b\pmod 7$.
We can simplify this equation as $5b\equiv 3\pmod 7$.
Similar to the previous step, we can try different values of $b$ from the set of positive integers less than $X$ and check if $5b$ gives a remainder of 3 when divided by 7.

Let's try $b=1$:
$5(1) = 5$, which gives a remainder of 5 when divided by 7.

Let's try $b=2$:
$5(2) = 10$, which gives a remainder of 3 when divided by 7.

Let's try $b=3$:
$5(3) = 15$, which gives a remainder of 1 when divided by 7.

Let's try $b=4$:
$5(4) = 20$, which gives a remainder of 6 when divided by 7.

Let's try $b=5$:
$5(5) = 25$, which gives a remainder of 4 when divided by 7.

Let's try $b=6$:
$5(6) = 30$, which gives a remainder of 2 when divided by 7.

From the above calculations, we can see that $b=3$ satisfies the third equation $6b\equiv 3+b\pmod 7$.

Now that we have found the values of $b$ and $c$, let's analyze the first equation $abc\equiv 1\pmod 7$.
Substituting the values $b=3$ and $c=6$ into the first equation, we have $a(3)(6)\equiv 1\pmod 7$.
Simplifying, we get $18a\equiv 1\pmod 7$.

To find the value of $a$, we can try different values from the set of positive integers less than $X$ and check if $18a$ gives a remainder of 1 when divided by 7.

Let's try $a=1$:
$18(1) = 18$, which gives a remainder of 4 when divided by 7.

Let's try $a=2$:
$18(2) = 36$, which gives a remainder of 1 when divided by 7.

From the above calculations, we can see that $a=2$ satisfies the first equation $abc\equiv 1\pmod 7$.

Now that we have found the values of $a$, $b$, and $c$, we can calculate the remainder when $a+b+c$ is divided by 7.
$a+b+c = 2 + 3 + 6 = 11$, which gives a remainder of 4 when divided by 7.

Therefore, if the answer to the above question is 4, the value of unknown variable $X$ is 7.
The answer is: 7
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
If $a,b,c$ are integers from the set of positive integers less than X such that \begin{align*}
abc&\equiv 1\pmod 7,\\
5c&\equiv 2\pmod 7,\\
6b&\equiv 3+b\pmod 7,
\end{align*}then what is the remainder when $a+b+c$ is divided by $7$?
If we know the answer to the above question is 4, what is the value of unknown variable X?

### Answer:
