
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
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \).

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