
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
Define a function $f(x)$ on $\mathbb{R}$ that satisfies:  
① For any $x, y \in \mathbb{R}$, $f(x+y) = f(x) + f(y)$  
② When $x > 0$, $f(x) < 0$, and $f(1) = -2$  
(1) Find the value of $f(0)$;  
(2) Determine the parity of the function $f(x)$;  
(3) Determine the monotonicity of the function $f(x)$;  
(4) Solve the inequality $f(x^2-2x) - f(x) \geq -8$.

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