Let $\odot$ be a binary operation on the natural numbers satisfying the properties that, for all $a, b,$ and $c$, $(a + b) \odot c = (a \odot c) + (b \odot c)$ and $a \odot (b + c) = (a \odot b) \odot c.$ Given that $5 \odot 5 = 160$, find the value of $7 \odot 7$.

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