
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  $ \omega$  be a circle with center O and radius 10, and let H be a point such that  $OH = 6$ . A point P is called snug if, for all triangles ABC with circumcircle ω and orthocenter  $H$ ,
we have that P lies on  $\triangle$ ABC or in the interior of  $\triangle$ ABC. Find the area of the region consisting of all snug points.

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