
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 $f(x) = ax^2 + 2bx + c$ $(x \in \mathbb{R}, a \neq 0)$.
(1) If $a=-1$, $c=0$, and the maximum value of $y=f(x)$ on the interval $[-2, 4]$ is $g(b)$, find $g(b)$.
(2) If $a > 0$, the function $f(x)$ is not monotonic on the interval $[-10, -2]$, and the range of $f(x)$ is $[0, +\infty)$, find the minimum value of $\frac{f(1)}{b - 2a}$.

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