
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 $y=f\left(x\right)$ that satisfies $f\left(0\right)=1$, and $f\left(x+1\right)=f\left(x\right)+2x$.  
$(1)$ Find the analytical expression of the function $f\left(x\right)$;  
$(2)$ If the function $g\left(x\right)=\left(t+1\right)x$, $t\in R$, and the function $h\left(x\right)=g\left(x\right)+f\left(x\right)$, find the minimum value of $h\left(x\right)$ on the interval $\left[-1,1\right]$.

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