
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 quadratic function $f(x) = ax^{2} + (a-1)x + a$.
1. If $a=2$, find the maximum value of the function $f(x)$ on the interval $[-1, 1]$.
2. Find the range of the real number $a$ such that the inequality $\frac{f(x)}{x} \geq 2$ holds for all $x \in [1, 2]$.
3. If the function $g(x) = f(x) + \frac{1-(a-1)x^{2}}{x}$ is increasing on the interval $(2, 3)$, find the range of the real number $a$.

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