
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]


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Given two tangent circles and a point \( P \) on their common tangent, which is perpendicular to the lines joining their centers, construct with ruler and compass all the circles that are tangent to these two circles and pass through point \( P \).

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