
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 $f(x) = ax^2 - |x| + 2a - 1$ (where $a$ is a real constant).  
(1) If $a=1$, find the intervals of monotonicity for $f(x)$.  
(2) If $a>0$, let the minimum value of $f(x)$ in the interval $[1, 2]$ be $g(a)$, find the expression for $g(a)$.

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