
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
Circle  $\Gamma$  with radius  $1$  is centered at point  $A$  on the circumference of circle  $\omega$  with radius  $7$ . Suppose that point  $P$  lies on  $\omega$  with  $AP=4$ . Determine the product of the distances from  $P$  to the two intersections of  $\omega$  and  $\Gamma$ .

*2018 CCA Math Bonanza Team Round #6*

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