
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]


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Designate the vertices of the square as \( A, B, C, D \) such that \( L \) lies on \( AB \), \( M \) lies on \( BC \), \( N \) lies on \( CD \), and \( K \) lies on \( AD \). Designate the intersection of \( KM \) and \( LN \) as \( O \). Designate the lengths of segments \( AL \) and \( AK \) as \( x \) and \( y \) respectively. Thus, by the Pythagorean theorem, \( 1 - x - y = \sqrt{x^2 + y^2} \). From this, squaring both sides yields \( 1 = 2x + 2y - 2xy \). Since \( AD \parallel LN \parallel BC \) and \( AB \parallel KM \parallel CD \), we have \( OL = y \) and \( OK = x \). Then \( OM = 1 - x \) and \( ON = 1 - y \). Therefore, \( NC =1 - x \) and \( MC = 1 - y \). Hence, the area of triangle \( NMC \) is \( \frac{1}{2} NC \cdot MC = \frac{1}{2}(1 - x)(1 - y) = \frac{1}{2}(1 - x - y + xy) = \frac{1}{4} \).

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