
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 $a \gt 1$, $m\in R$, $f(x)=\frac{{a}^{m}}{x}$. When $x\in \left[a,2a\right]$, the range of $f\left(x\right)$ is $[a^{2}$, $a^{3}]$.  
$(1)$ Find the value of $a$;  
$(2)$ If there exists a real number $t$ such that $\left(x+t\right)^{2}+2\left(x+t\right)\leqslant \left(a+1\right)x$ holds for all $x\in \left[0,s\right]$, find the range of the real number $s$.

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