Good functions

Number Theory Level pending

Given an integer \(n\ge 2\), a function \(f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}\) is called good, if for any integer \(k,1\le k\le n-1\) there exists an integer \(j(k)\) such that for every integer \(m\) we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of good functions when \(n=12\).

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