
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
Consider the set \( N = \{1,2,3, \ldots\} \) of natural numbers and the function \( f \): \( N \rightarrow N \) that satisfies the following two conditions:

a) \( f(f(n))=n \) for all \( n \in N \).

b) \( f(f(n)+1)=\left\{\begin{array}{cc}n-1, & \text { if } n \text { is even; } \\ n+3, & \text { if } n \text { is odd. }\end{array}\right. \)

Determine the value of \( f(n) \) for each \( n \in N \), noting that \( f \) is bijective and since \( f(f(n)+1) \neq 2 \), it must be that \( f(1)=2 \).

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