
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 area  $5$  and  $BC = 10.$  Let  $E$  and  $F$  be the midpoints of sides  $AC$  and  $AB$  respectively, and let  $BE$  and  $CF$  intersect at  $G.$  Suppose that quadrilateral  $AEGF$  can be inscribed in a circle. Determine the value of  $AB^2+AC^2.$ *Proposed by Ray Li*

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