
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
Suppose that  $n$  is a positive integer and that  $a$  is the integer equal to  $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$ If the sum of the digits of  $a$  is 567, what is the value of  $n$ ?

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