
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
Consider an isosceles right triangle with a hypotenuse of $60 \text{ cm}$. There are two natural ways to inscribe a square in such a triangle. If a square is inscribed in one of these ways, its corners touch the midpoint of one leg, the vertex opposite this leg, and the midpoint of the hypotenuse. You are given that the side of the square inscribed in the triangle as described results in an area of $400 \text{cm}^2$. Calculate the area of another square inscribed in the triangle, touching the midpoint of the other leg and both endpoints of the hypotenuse.

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