
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 $g\left(x\right)=te^{2x}+\left(t+2\right)e^{x}-1$, where $t\in \mathbb{R}$ and $e$ is the base of the natural logarithm.<br/>$(1)$ If the function $g\left(x\right)$ is monotonically decreasing on $\left[0,+\infty \right)$, find the range of real numbers for $t$;<br/>$(2)$ If $t$ is a non-negative real number and the function $f\left(x\right)=g\left(x\right)-4e^{x}-x+1$ has a unique zero on $R$, find the value of $t$.

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