
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 function $y=f(x)$ be defined on $\mathbb{R}$ and satisfy $f(x-y)=f(x)-f(y)$, with $f(2)=1$. If $f(x)>0$ for $x>0$,  
(1) find the value of $f(0)$;  
(2) determine the monotonicity of the function $f(x)$ and provide a proof;  
(3) if $f(x)+f(x+2)<2$, find the range of $x$.

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