
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 $a > 0$, the function $f(x) = x^2 + a|\ln x - 1|$  
(1) When $a = 2$, find the intervals of monotonic increase for the function $f(x)$.  
(2) If $x \in [1, +\infty)$, the inequality $f(x) \geq a$ always holds, find the range of values for the real number $a$.

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