Question 14
For integers $a$ and $b,$ let $a \circ b = a - b$ if $a$ is odd and $b$ is even, and $a \circ b = a + b$ otherwise. Find the number of sequences $a_1,$ $a_2,$ $a_3,$ $\ldots,$ $a_n$ of positive integers such that $$
a_1 + a_2 + a_3 + \cdots + a_n = 12
$$ and $$
a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_n = 0,
$$ where the operations are performed from left to right; that is, $a_1 \circ a_2 \circ a_3$ means $(a_1 \circ a_2) \circ a_3.$
a_1 + a_2 + a_3 + \cdots + a_n = 12
$$ and $$
a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_n = 0,
$$ where the operations are performed from left to right; that is, $a_1 \circ a_2 \circ a_3$ means $(a_1 \circ a_2) \circ a_3.$