
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 \(C\) divides diameter \(AB\) in the ratio \(AC:BC = 2:1\). A point \(P\) is selected on the circle. Determine the possible values that the ratio \(\tan \angle PAC: \tan \angle APC\) can take. Specify the smallest such value.

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