
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
Professor Rakbrain once suggested to his friends one morning to find at least three integers that form a geometric progression starting with 1, and whose sum is a perfect square. (For example, $1+2+4+8+16+32=63$. However, the last number falls short by one to be a perfect square. Only two solutions in integers are known to me; finding both is not difficult.)

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