
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 $f(x) = ax^2 + bx + c$, where $a\in\mathbb{N}^*$, $b\in\mathbb{N}$, and $c\in\mathbb{Z}$.  
(1) If $b>2a$ and $f(\sin x)$ (where $x\in\mathbb{R}$) has a maximum value of 2 and a minimum value of -4, find the minimum value of the function $f(x)$.  
(2) If the inequality $4x \leq f(x) \leq 2(x^2 + 1)$ always holds for any real number $x$, and there exists an $x_0$ such that $f(x_0) < 2(x_0^2 + 1)$, determine the value of $c$.

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