
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 that $a$ is a real number, the function $f(x) = x^2 - 2a\ln x$,
1. Find the monotonic intervals of the function $f(x)$.
2. Find the minimum value $g(a)$ of the function $f(x)$ on $[1, +\infty)$.
3. If $a > 0$, find the value of $a$ such that the equation $f(x) = 2ax$ has a unique solution.

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