
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 $f(x) = x|x-a|+2x$.  

(1) If the function $f(x)$ is an increasing function on $\mathbb{R}$, find the range of the real number $a$;  
(2) Find all real numbers $a$, such that for any $x \in [1,2]$, the graph of the function $f(x)$ always lies below the graph of the function $g(x) = 2x+1$;  
(3) If there exists $a \in [-4,4]$ such that the equation $f(x) = tf(a)$ has three distinct real roots with respect to $x$, find the range of the real number $t$.

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