
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
Point  $P$  is  $\sqrt3$  units away from plane  $A$ . Let  $Q$  be a region of  $A$  such that every line through  $P$  that intersects  $A$  in  $Q$  intersects  $A$  at an angle between  $30^o$  and  $60^o$  . What is the largest possible area of  $Q$ ?

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