
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
Petya and Vasya have identical paper right triangles with legs \(a\) and \(b\). The boys want to cut out the largest possible square such that one vertex of Petya's square coincides with the right angle vertex of the triangle, and one side of Vasya's square lies along the hypotenuse.
a) Find the dimensions of both squares.
b) Will Petya's square always be larger than Vasya's square (i.e., for any legs \(a\) and \(b\))?

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