
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 the function $f(x)=e^{ax}\cdot \sin x$.  
$(1)$ When $a=1$, find the interval where the function $f(x)$ is monotonically increasing on the interval $\left[0,2\pi \right]$.  
$(2)$ When $a\geqslant 1$, if $\forall x\in[0,\frac{π}{2}]$, $f(x)\leqslant bx$ holds, find the minimum value of $b-e^{2}a$.

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