
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
Two infinite geometric progressions are given with a common ratio \( |q| < 1 \), differing only in the sign of their common ratios. Their sums are \( S_{1} \) and \( S_{2} \). Find the sum of the infinite geometric progression formed from the squares of the terms of either of the given progressions.

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