
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
Let \( P_1 \) and \( P_2 \) be two points on a plane. Define \( P_{2k+1} \) as the point symmetric to \( P_{2k} \) with respect to \( P_1 \), and \( P_{2k+2} \) as the point symmetric to \( P_{2k+1} \) with respect to \( P_2 \), where \( k \in \mathbf{Z}_{+} \). If the distance \( \left| P_1 P_2 \right| = 1 \), find the distance \( \left| P_{2017} P_{2018} \right| \).

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