
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 $f(x) = ax^2 - x + 2a - 1$ ($a > 0$).

(1) If $f(x)$ is monotonically increasing in the interval $[1, 2]$, find the range of values for $a$.

(2) Let the minimum value of the function $f(x)$ in the interval $[1, 2]$ be $g(a)$, find the expression for $g(a)$.

(3) Let the function $$h(x) = \left( \frac {1}{2} \right)^{x} + \log_{2} \frac {1}{x+1}$$, if for any $x_1, x_2 \in [1, 2]$, the inequality $f(x_1) \geq h(x_2)$ 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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