
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given the set \( P = \{ x \mid x = 7^3 + a \times 7^2 + b \times 7 + c \} \), where \( a \), \( b \), and \( c \) are positive integers not exceeding 6, if \( x_1, x_2, \ldots, x_n \) are elements of set \( P \) forming an arithmetic sequence, find the maximum value of \( n \).

Present the answer in LaTex format: \boxed{Your answer}
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