
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 ways are there to choose 2010 functions \( f_{1}, \ldots, f_{2010} \) from \(\{0,1\}\) to \(\{0,1\}\) such that \( f_{2010} \circ f_{2009} \circ \cdots \circ f_{1} \) is constant? Note: a function \( g \) is constant if \( g(a)=g(b) \) for all \( a, b \) in the domain of \( g \).

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