
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 $f(x) = x|x-a|+bx$ where $a,b \in \mathbb{R}$.
(1) When $b = -1$, the function $f(x)$ has exactly two distinct zeros. Find the value(s) of the real number $a$.
(2) When $b = 1$,
(a) If for all $x \in [1,3]$, we have $\dfrac{f(x)}{x} \leqslant 2\sqrt{x+1}$, find the range of values for $a$.
(b) If $a > 0$, find the maximum value of the function $f(x)$ on the interval $[0,2]$, denoted as $g(a)$.

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