
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
**O**n a blackboard a stranger writes the values of  $s_7(n)^2$  for  $n=0,1,...,7^{20}-1$ , where  $s_7(n)$  denotes the sum of digits of  $n$  in base  $7$ . Compute the average value of all the numbers on the board.

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