
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 $a$, $b$ are real numbers, the function $f(x) = x^2 + ax + 1$, and the function $y = f(x + 1)$ is an even function. The function $g(x) = -b \cdot f(f(x + 1)) + (3b - 1) \cdot f(x + 1) + 2$ is decreasing on the interval $(-\infty, -2]$ and increasing on the interval $(-2, 0)$.  
(1) Find the expression for the function $f(x)$.  
(2) Find the value of the real number $b$.  
(3) Let $h(x) = f(x + 1) - 2qx + 1 + 2q$, is there a real number $q$ such that $h(x)$ has a minimum value of $-2$ on the interval $[0, 2]$? If so, find the value of $q$; if not, explain why.

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