
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 an isosceles right triangle with legs of length $a$, find the limit of the sum of the perimeters of all triangles formed by joining the midpoints of the legs of the previous triangle indefinitely.

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