
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 \(ABC\) be a triangle with \(AB=13\), \(BC=14\), and \(CA=15\). We construct an isosceles right triangle \(ACD\) with \(\angle ADC=90^\circ\), where \(D\) and \(B\) are on the same side of line \(AC\), and let lines \(AD\) and \(CB\) meet at \(F\). Similarly, we construct an isosceles right triangle \(BCE\) with \(\angle BEC=90^\circ\), where \(E\) and \(A\) are on the same side of line \(BC\), and let lines \(BE\) and \(CA\) meet at \(G\). Find \(\cos \angle AGF\).

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