
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 trapezoid \(AEFG\) (\(EF \parallel AG\)) is positioned inside a square \(ABCD\) with a side length of 14, such that points \(E\), \(F\), and \(G\) lie on sides \(AB\), \(BC\), and \(CD\) respectively. The diagonals \(AF\) and \(EG\) are perpendicular, and \(EG = 10\sqrt{2}\). Find the perimeter of the trapezoid.

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