
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 $f(x) = x^2 - x|x-a| - 3a$ where $a > 0$,
1. If $a = 1$, determine the intervals of monotonicity for $f(x)$.
2. Find the maximum and minimum values of the function on the interval $x \in [0,3]$.
3. When $a \in (0,3)$, if the function $f(x)$ has exactly two distinct zeros denoted by $x_1$ and $x_2$, find the range of $|\frac{1}{x_1} - \frac{1}{x_2}|$.

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