For every positive integer $n, \cos (nx)$ can be written as a polynomial in $\cos (x)$ with integral coefficients. We call it $P_{n}(y)$ where $y=\cos (x)$. For example, we have $P_{1}(y)=y, P_{2}(y)=2y^{2}-1$.

Denote $S (n)$ to be the sum of absolute values of the coefficients of $P_{n}(y)$. For example, $S(1)=1, S (2)=3$.

Find the smallest positive value of $A$ such that for all integers $n> 0$,

$S(n) \equiv S(n+A) \pmod{100}.$

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