
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 function $f(x) = ax^2 - bx + 1$ $(a, b \in \mathbb{R})$.  
(1) If the range of the function $f(x)$ is $\left[\frac{3}{4}, +\infty \right)$, and $f(x+1) = f(-x)$, find the explicit formula for $f(x)$.  
(2) Let $b = a + 1$, when $0 \leq a \leq 1$, for every $x \in [0, 2]$, $m \geq |f(x)|$ always holds true, determine the minimum value of $m$.

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