
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 $f(x)$ that satisfies: For all $x\in \mathbb{R}$, $f(x+1)-f(x)=2x-2$, and $f(1)=-2$.  
$(1)$ Find the analytical expression of the function $f(x)$;  
$(2)$ If the only integer solution that satisfies the inequality system $\left\{\begin{array}{l}f(x)>0\\ f(x+t)<0\end{array}\right.$ is one, find the range of negative real number $t$.

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