We are given the equations $a - 7 = N$, $b + 7 = N$, and $7c = N$.
Since the sum of $a$, $b$, and $c$ is 60, we have $a + b + c = 60$.
Substituting $N$ for $a - 7$, $b + 7$, and $7c$, we get $(N + 7) + (N - 7) + \frac{N}{7} = 60$.
Combining like terms, we have $2N + \frac{N}{7} = 60$.
Multiplying both sides by 7 to get rid of the fraction, we have $14N + N = 420$.
Combining like terms, we get $15N = 420$, so $N = \boxed{28}$.
The answer is: 28
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
The sum of three numbers $a, b$ and $c$ is 60. If we decrease $a$ by 7, we get the value $N$. If we increase $b$ by 7, we get the value $N$. If we multiply $c$ by 7, we also get the value $N$. What is the value of $N$?

### Answer:
