
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
Isabella has a sheet of paper in the shape of a right triangle with sides of length 3, 4, and 5. She cuts the paper into two pieces along the altitude to the hypotenuse, and randomly picks one of the two pieces to discard. She then repeats the process with the other piece (since it is also in the shape of a right triangle), cutting it along the altitude to its hypotenuse and randomly discarding one of the two pieces once again, and continues doing this forever. As the number of iterations of this process approaches infinity, the total length of the cuts made in the paper approaches a real number  $l$ . Compute  $[\mathbb{E}(l)]^2$ , that is, the square of the expected value of  $l$ .

*Proposed by Matthew Kroesche*

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