
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
A square \( KLMN \) with side length 1 is inscribed in triangle \( ABC \): points \( K \) and \( L \) lie on side \( AC \), and points \( M \) and \( N \) lie on sides \( AB \) and \( BC \) respectively. The area of the square is half the area of the triangle. Find the length of the altitude \( BH \) of triangle \(ABC\).

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