
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 a function $y=f(x)$ defined on $\mathbb{R}$ satisfies: for all $x, y \in \mathbb{R}$, $f(x+y)=f(x)+f(y)-3$, and when $x>0$, $f(x)<3$.  
(1) Find the value of $f(0)$;  
(2) Determine the monotonicity of $f(x)$ on $\mathbb{R}$ and provide a proof;  
(3) If the inequality $f((t-2)|x-4|)+3>f(t^2+8)+f(5-4t)$ holds for all $t \in (2, 4)$, find the range of the real number $x$.

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