
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 multiplicative group  $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$  of all the roots of unity of degree  $2006^k$  for all positive integers  $k$ .

Find the number of homomorphisms  $f:A\to A$  that satisfy  $f(f(x))=f(x)$  for all elements  $x\in A$ .

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