
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 monotonic function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x+y) = f(x) + f(y)$ for any $x, y$, and $f(1) = 1$.  
(1) Find the value of $f(0)$ and determine the parity of $f(x)$.  
(2) Solve the inequality with respect to $x$: $f(x-x^2+2) + f(2x) + 2 < 0$.

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