
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  $\triangle ABC$  with incenter  $I$  and  $AB=5$ ,  $BC=12$ ,  $CA=13$ . A circle  $\omega$  centered at  $I$  intersects  $ABC$  at  $6$  points. The green marked angles sum to  $180^\circ.$  Find  $\omega$ 's area divided by  $\pi.$ 

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