
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
Let the set $M = \{-2, 0, 1\}$ and $N = \{1, 2, 3, 4, 5\}$. A function $f: M \rightarrow N$ is defined such that for any $x \in M$, $x + f(x) + x f(x)$ is an odd number. Determine how many such functions $f$ exist.

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