
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 $f(x)=|x|(x-a)$, where $a$ is a real number.
1. If the function $f(x)$ is an odd function, find the value of the real number $a$.
2. If the function $f(x)$ is increasing on the interval $[0,2]$, find the range of values for the real number $a$.
3. Determine whether there exists a real number $a(a < 0)$ such that the maximum value of $f(x)$ on the closed interval $[-1, \frac{1}{2}]$ is $2$. If it exists, find the value of $a$; if not, explain the reason.

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