
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
The function \( f(x) \) has the following properties:
(i) \( f(0) = 0 \) and \( f(1) = 1 \).
(ii) There are infinitely many positive integers \( n \) such that if the equation
\[ f\left(x_{1}\right) + f\left(x_{2}\right) + \cdots + f\left(x_{s}\right) = n \]
has integer solutions \( x_{1}, x_{2}, \cdots , x_{s} \), then \( s \geq 2^{k} - 1 \).

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