Find an integer $a$, with $0 \leq a<35$, such that the system of equations \[ \left\{\begin{array}{l} {[a][x]+[13][y]=[27]} \\ {[32][x]+[18][y]=[32]} \end{array}\right. \] has exactly 7 solutions in $\mathbb{Z}_{35}$.

Step 1 ：The system of equations is in the form of a linear Diophantine equation. We need to find an integer 'a' such that the system has exactly 7 solutions.

Step 2 ：We can solve this problem by iterating over all possible values of 'a' and checking the number of solutions for each value.

Step 3 ：We can use the Extended Euclidean Algorithm to find the solutions of the equations.

Step 4 ：After performing the above steps, we find that the integer 'a' that satisfies the given conditions is 13.

Step 5 ：Final Answer: The integer \(a\) that satisfies the given conditions is \(\boxed{13}\).