
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 $f(x) = x|x^2 - a|$, where $a \in \mathbb{R}$.

1. Find the equation of the tangent line to the curve $y = f(x)$ at $x = 1$ when $a = 2$.
2. Discuss the monotonicity of the function $f(x)$ when $a > 0$.
3. Find the range of $a$ such that for any $x_1, x_2 \in [0, 1]$, the inequality $|f(x_1) - f(x_2)| < \frac{\sqrt{2}}{2}$ holds when $a > 0$.

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