
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  $\mathbb{N}_0=\{0,1,2 \cdots \}$ . Does there exist a function  $f: \mathbb{N}__0 \to \mathbb{N}_0$  such that:
\[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0  \]
where we define:  $f^1(n)=f(n)$  and  $f^{k+1}(n)=f(f^k(n))$ ,  $\forall k \in \mathbb{N}_0$ ?

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