
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
On the ground, there are three points: \( A \), \( B \), and \( C \). A frog is at point \( P \) on the ground, which is 0.27 meters away from point \( C \). The frog's first jump is a "symmetric jump" from \( P \) to the symmetric point \( P_{1} \) with respect to point \( A \). The second jump is a symmetric jump from \( P_{1} \) with respect to point \( B \) to reach \( P_{2} \). The third jump is a symmetric jump from \( P_{2} \) with respect to point \( C \) to reach \( P_{3} \). The fourth jump is a symmetric jump from \( P_{3} \) with respect to point \( A \) to reach \( P_{4} \), and so on, jumping in this way continuously. After the 2009th symmetric jump, the frog reaches \( P_{2009} \). What is the distance between \( P_{2009} \) and the starting point \( P \) in centimeters?

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