Let $a$, $b$, and $c$ be real numbers such that $0 \le a, b, c \le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$.

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