
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 quadratic function $f(x) = ax^2 + bx + c$ where $a, b, c \in \mathbb{R}$. If for all real numbers $x$, the inequality $x^2 - 2x + 2 \leq f(x) \leq 2x^2 - 4x + 3$ always holds, find the value of $a + b + c$.

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