Let \( a_n \) be a sequence defined recursively by \( a_0 = a_1 = \dots = a_5 = 1 \) and

\[

a_n = \frac{a_{n-1} a_{n-5} + a_{n-2} a_{n-4} + a_{n-3}^2}{a_{n-6}}

\]

for \( n \geq 6 \). Prove or disprove that every \( a_n \) is an integer.

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