Example 1 Find all positive integer solutions $(x, y)$ for $\left|12^{x}-5^{y}\right|=7$.

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Solution: First, $(x, y)=(1,1)$ is a solution, and we assume $x>1$, $y>1$ below.
(1) If $12^{x}-5^{y}=7$, then $-(-1)^{y} \equiv 1$ $(\bmod 3), \therefore y$ is odd.
$$
\because 12^{x} \equiv 0(\bmod 8), \therefore 5^{y} \equiv 1(\bmod 8),
$$

but $5^{y} \equiv 5^{2 k+1} \equiv 5(\bmod 8)$, contradiction!
(2) If $5^{y}-12^{x}=7$, then $1 \equiv 3(\bmod 4)$, contradiction!
Therefore, the equation has a unique solution $(x, y)=(1,1)$.
It can be seen that choosing an appropriate modulus (such as $\bmod 3$, $\bmod 8$ above) facilitates simplifying the equation and revealing the nature of the solution. The choice of modulus depends on the specific equation, generally aiming to facilitate calculation and to result in remainders of $0, \pm 1$ as much as possible, which helps in finding the relationship between variables. This method is commonly used in handling exponential equations.