
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 $f(x) = x|x-a| + 2x$, where $a \in \mathbb{R}$.
1. If the function $f(x)$ is increasing on $\mathbb{R}$, find the range of values for $a$.
2. If there exists $a \in [-2, 4]$ such that the equation $f(x) = bf(a)$ has three distinct real solutions, find the range of values for the real number $b$.

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