
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
How many non-congruent squares can be drawn, such that their vertices are lattice points on the 6 by 6 grid of lattice points shown? Additionally, consider squares whose vertices lie on any hexagon that can be inscribed within the grid by connecting lattice points.

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