The increasing sequence of positive integers $a_1,$ $a_2,$ $a_3,$ $\dots$ has the property that
\[a_{n + 2} = a_{n + 1} + a_n\]for all $n \ge 1.$  If $a_7 = 120,$ then find $a_8.$

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