
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
Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let  $\omega_1,\omega_2,\omega_3$  be three circles of radii  $3, 5,$  and  $7$ , respectively. Suppose they are arranged in such a way that the common chord of  $\omega_1$  and  $\omega_2$  is a diameter of  $\omega_1$ , the common chord of  $\omega_1$  and  $\omega_3$  is a diameter of  $\omega_1$ , and the common chord of  $\omega_2$  and  $\omega_3$  is a diameter of  $\omega_2$ . Compute the square of the area of the triangle formed by the centers of the three circles.


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