
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 $f\left(x\right)=x^{2}+t\sin x$.<br/>$(1)$ If the slope of the tangent line at $x=2\pi$ is $2\pi -2$, find the range of $m$ for which $f\left(x\right)\geqslant 2x+m$ holds for $\left[0,+\infty \right)$;<br/>$(2)$ When $x\in \left(0,\pi \right)$, the equation $f\left(x\right)=x^{2}+\ln \left(x+1\right)-x$ has a unique root. Find the range of $t$.

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