
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
If \( a \) and \( b \) are positive integers that can each be written as a sum of two squares, then \( ab \) is also a sum of two squares. Find the smallest positive integer \( c \) such that \( c = ab \), where \( a = x^3 + y^3 \) and \( b = x^3 + y^3 \) each have solutions in integers \( (x, y) \), but \( c = x^3 + y^3 \) does not.

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