
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 $f\left(x\right)=ax^{2}+bx+c$.  
$(1)$ If $f\left(-1\right)=0$ and $f\left(x+1\right)-f\left(x\right)=2x$, find the analytical expression of the function $f\left(x\right)$.  
$(2)$ Determine whether there exist $a$, $b$, $c \in \mathbb{R}$ that satisfy the following three conditions for the function $f\left(x\right)$:  
① The range is $\left[0,+\infty \right)$.  
② For all $x \in \mathbb{R}$, $f\left(x-4\right)=f\left(2-x\right)$.  
③ For all $x \in \mathbb{R}$, $0 \leq f(x)-x \leq \frac{1}{2}(x-1)^2$. If they exist, find the values of $a$, $b$, $c$; if not, explain the reason.

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