Consider a sequence defined by the recurrence relation:
\[a_n = 2a_{n-1} + 3a_{n-2}\]
with the initial conditions \(a_0 = 1\) and \(a_1 = 2\). Determine the 5th term of the sequence, \(a_5\). Let's think step by step and output the final answer within \boxed{}.
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