Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\ldots,n\}$
such that
$|i-j|=1$  implies  $|\pi(i)-\pi(j)|\leq2$
for all $i,j$ in $\{1,2,\ldots,n\}.$ Show that for $n\geq2$,the quantity
$$P_{n+5}-P_{n+4}-P_{n+3}+P_n$$
does not depend on $n$, and find its value.

Let's think step by step and output the final answer within \boxed{}.
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