
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 \( S \) be the set of \( 3^4 \) points in four-dimensional space where each coordinate is in \( \{-1, 0, 1\} \). Let \( N \) be the number of sequences of points \( P_1, P_2, \ldots, P_{2020} \) in \( S \) such that \( P_i P_{i+1}=2 \) for all \( 1 \leq i \leq 2020 \) and \( P_1 = (0,0,0,0) \). (Here \( P_{2021} = P_1 \).) Find the largest integer \( n \) such that \( 2^n \) divides \( N \).

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