
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
 In triangle  $ABC$ ,  $AB = 13$ ,  $BC = 14$ , and  $CA = 15$ . Let  $M$  be the midpoint of side  $AB$ ,  $G$  be the centroid of  $\triangle ABC$ , and  $E$  be the foot of the altitude from  $A$  to  $BC$ . Compute the area of quadrilateral  $GAME$ .

*Proposed by Evin Liang*

<details><summary>Solution</summary>*Solution*.  $\boxed{23}$ Use coordinates with  $A = (0,12)$ ,  $B = (5,0)$ , and  $C = (-9,0)$ . Then  $M = \left(\dfrac{5}{2},6\right)$  and  $E = (0,0)$ . By shoelace, the area of  $GAME$  is  $\boxed{23}$ .</details>

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