
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
The cross-section of the attic of a building is a symmetric pentagon with a vertical axis, which can be divided into a rectangle and an isosceles triangle standing on it. The common base of the two shapes is $2a$, and their heights are $b$ and $m$, respectively. For the purpose of determining the cross-section of the living room to be built, find the base, height, and area of the largest rectangle that can be inscribed in the pentagon. Construct the vertices of the rectangle.

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