
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 $f(x)=e^{x}-mx^{3}$ ($m$ is a nonzero constant).  
$(1)$ If the function $f(x)$ is increasing on $(0,+\infty)$, find the range of real numbers for $m$.  
$(2)$ If $f_{n+1}(x)$ ($n\in \mathbb{N}$) represents the derivative of $f_{n}(x)$, where $f_{0}(x)=f(x)$, and when $m=1$, let $g_{n}(x)=f_{2}(x)+f_{3}(x)+\cdots +f_{n}(x)$ ($n\geqslant 2, n\in \mathbb{N}$). If the minimum value of $y=g_{n}(x)$ is always greater than zero, find the minimum value of $n$.

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