
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 $f(x)=(a+1)x^{2}+(x-4)|x-a|-x$, where $a\in \mathbb{R}$.<br/>$(1)$ When $a=0$, find the monotonic intervals of $f(x)$;<br/>$(2)$ If for any $x_{1}$, $x_{2}\in [0,5]$, and $x_{1}\neq x_{2}$, $\frac{{f(x_{1})-f(x_{2})}}{{x_{1}^2-x_{2}^2}} > a$ holds, find the range of real number $a$.

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