You are given an integer k$k$. Find the largest integer x$x$, where 1≤x<k$1\le x<k$, such that x!+(x−1)!†$x!+(x-1){!}^{†}$ is a multiple of ‡${}^{‡}$ k$k$, or determine that no such x$x$ exists.

†${}^{†}$ y!$y!$ denotes the factorial of y$y$, which is defined recursively as y!=y⋅(y−1)!$y!=y\cdot (y-1)!$ for y≥1$y\ge 1$ with the base case of 0!=1$0!=1$. For example, 5!=5⋅4⋅3⋅2⋅1⋅0!=120$5!=5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 0!=120$.

‡${}^{‡}$ If a$a$ and b$b$ are integers, then a$a$ is a multiple of b$b$ if there exists an integer c$c$ such that a=b⋅c$a=b\cdot c$. For example, 10$10$ is a multiple of 5$5$ but 9$9$ is not a multiple of 6$6$.