
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
Evaluate the limit of the sum of an infinite geometric progression starting from a term \( a + t \) (where \( t \) is a constant), and then having terms that follow the common ratio \( r + k \) (\( |r + k| < 1 \)). Calculate the limit of the sum of their squares.
**A**. \( \frac{(a + t)^2}{1 - (r + k)^2} \)  
**B**. \( \frac{(a + t)^2}{1 + (r + k)^2} \)  
**C**. \( \frac{(a + t)^2}{(1 - (r + k))^2} \)  
**D**. \( \frac{4(a + t)^2}{1 + (r + k)^2} \)  
**E**. None of these

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