
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
Consider the 81 points where the 9x9 grid lines of a chessboard intersect, and let's call these points network points. How many squares are there such that all four of their vertices are network points? How many remain if the center point of the board cannot be a vertex of the square?

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