
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 $a > 0$, let $f(x) = x^2 + a|\ln x - 1|$.
(1) When $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at $x=1$;
(2) When $a=3$, determine the monotonicity of the function $f(x)$;
(3) When $x \in [1, +\infty)$, find the minimum value of the function $f(x)$.

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