Suppose a positive integer \( n \) can be expressed in base \( 7 \) as \(\overline{ABC}_7\) and in base \( 11 \) as \(\overline{CBA}_{11}\), where \( A, B, \) and \( C \) are digits. Find the largest possible value of \( n \) in base \( 10 \). Let's think step by step and output the final answer within \boxed{}.
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