
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
Consider a quadratic function $f(x) = ax^2 + bx + c$ where $a, b, c \in \mathbb{R}$. The function satisfies the following conditions:
1. For all $x \in \mathbb{R}$, the minimum value of $f(x)$ is $0$, and its graph is symmetric about the line $x = -1$.
2. For $x \in (0, 5)$, it holds that $x \leq f(x) \leq 2|x - 1| + 1$.

a. Find the value of $f(1)$.
b. Determine the explicit formula of the function $f(x)$.
c. If $|f(x) - x| \leq 1$ is always true for $x \in [m - 1, m]$, find the range of values for the real number $m$.

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