
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 an acute triangle  $ABC$ , the points  $H$ ,  $G$ , and  $M$  are located on  $BC$  in such a way that  $AH$ ,  $AG$ , and  $AM$  are the height, angle bisector, and median of the triangle, respectively. It is known that  $HG=GM$ ,  $AB=10$ , and  $AC=14$ . Find the area of triangle  $ABC$ .

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