
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
Let \( EFGH \) be an isosceles trapezoid with bases \( EF = 102 \) and \( GH = 26 \). Suppose \( EG = FH = y \) and a circle with center on \( \overline{EF} \) is tangent to segments \( \overline{EG} \) and \( \overline{FH} \). If \( n \) is the smallest possible value of \( y \), then find \( n^2 \).

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