AIME 2026b Problem 14

For integers $a$ and $b$, let $a\circ b = a-b$ if $a$ is odd and $b$ is even, and $a+b$ otherwise. Find the number of sequences $a_1,a_2,a_3,\dots,a_n$ of positive integers such that $$a_1+a_2+a_3+\dots + a_n = 12\text{ and }a_1\circ a_2\circ a_3\circ\dots\circ a_n = 0,$$ where the operations are performed from left to right; that is, $a_1\circ a_2\circ a_3$ means $(a_1\circ a_2)\circ a_3$.