
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 $f(x) = ax^2 + (b - 8)x - a - ab$ ($a \neq 0$), when $x \in (-3,2)$, $f(x) > 0$;  
when $x \in (-\infty, -3) \cup (2, +\infty)$, $f(x) < 0$.  
(1) Find the range of $f(x)$ in $[0,1]$.  
(2) For what value of $c$ does the inequality $ax^2 + bx + c \leq 0$ always hold in $[1,4]$.

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