
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
A function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is essentially increasing if \( f(s) \leq f(t) \) holds whenever \( s \leq t \) are real numbers such that \( f(s) \neq 0 \) and \( f(t) \neq 0 \).

Find the smallest integer \( k \) such that for any 2022 real numbers \( x_{1}, x_{2}, \ldots, x_{2022} \), there exist \( k \) essentially increasing functions \( f_{1}, \ldots, f_{k} \) such that
\[ f_{1}(n) + f_{2}(n) + \cdots + f_{k}(n) = x_{n} \quad \text{for every } n = 1, 2, \ldots, 2022. \]

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