
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
Given the terms of two independent geometric sequences $a, ar, ar^2, ar^3, \ldots$ and $b, br, br^2, br^3, \ldots$ where $r$ is the common ratio, and the series for the sum of their terms is '$a+b$, $a+b$, $(a+b)r$, $(a+b)r^2, \ldots$', and $|r| < 1$, find the limit of the sum of the squares of the terms of the new combined sequence.

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