Sokal da is trying to find the largest positive integer $n$ such that when $n$ is transformed to base-7, it appears as twice its value in base-10. For example, $156$ is such a number because $156_{10} = 312_7$ and $312 = 2 \times 156$. Determine Sokal da's number.

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