
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
Let $a$, $b$, $c$ be real numbers, and $a\neq 0$. Given a quadratic function $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=2$ and $f\left(x+1\right)-f\left(x\right)=-2x-1$.  
$(1)$ Find the analytical expression of the function $f\left(x\right)$.  
$(2)$ Let $t\in \mathbb{R}$. Find the maximum value of the function $f\left(x\right)$ denoted by $g\left(t\right)$ when $x\in \left[t,t+2\right]$.

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