
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 $f(x)=x|x-a|$,  
$(1)$ If the function $y=f(x)+x$ is increasing on $\mathbb{R}$, find the range of the real number $a$;  
$(2)$ If for any $x \in [1,2]$, the graph of the function $f(x)$ is always below the graph of $y=1$, find the range of the real number $a$;  
$(3)$ Suppose $a \geqslant 2$, find the range of $f(x)$ in the interval $[2,4]$.

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