
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
How many permutations \(\left(i_{1}, i_{2}, \ldots, i_{n}\right)\) of the numbers \(1, 2, \ldots, n\) are there such that for every \(1 \leq k \leq n\),

$$
\left|i_{k}-k\right| \leq 1
$$

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