Let $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ be sequences defined by the recurrence relations:
\[
a_{n+2} = a_{n+1} + a_n, \quad n = 0, 1, \ldots, \quad a_0 = 1, \quad a_1 = 2,\nb_{n+2} = b_{n+1} + b_n, \quad n = 0, 1, \ldots, \quad b_0 = 2, \quad b_1 = 1.
\]
Determine how many integers the sequences $(a_n)$ and $(b_n)$ have in common.

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