
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 a rectangle with length $l$ and width $w$. A triangle is constructed on the diagonal of the rectangle as the base such that its area equals that of the rectangle. Find the length of the altitude of the triangle drawn to the diagonal.
**A)** $\frac{2l}{\sqrt{l^2 + w^2}}$  
**B)** $\frac{2w}{\sqrt{l^2 + w^2}}$  
**C)** $\frac{2lw}{l + w}$  
**D)** $\frac{2lw}{\sqrt{l^2 + w^2}}$  
**E)** $\frac{l + w}{\sqrt{l^2 + w^2}}$

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