
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  $AB = 26$ ,  $BC = 51$ , and  $CA = 73$ , and let  $O$  be an arbitrary point in the interior of  $\vartriangle ABC$ . Lines  $\ell_1$ ,  $\ell_2$ , and  $\ell_3$  pass through  $O$  and are parallel to  $\overline{AB}$ ,  $\overline{BC}$ , and  $\overline{CA}$ , respectively. The intersections of  $\ell_1$ ,  $\ell_2$ , and  $\ell_3$  and the sides of  $\vartriangle ABC$  form a hexagon whose area is  $A$ . Compute the minimum value of  $A$ .

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