
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
Given the function $y=f(x)$, for any real numbers $x_1$, $x_2$, it holds that $f(x_1+x_2)=f(x_1)+f(x_2)$, and when $x>0$, $f(x)<0$  
(1) Find $f(0)$.  
(2) Determine the monotonicity of the function $y=f(x)$ and provide a proof.  
(3) If $f(x)+f(2-3x)<0$, find the range of $x$.

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