
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
If the function $y=f(x)$ satisfies that for any $x \in A$ in its domain, $e^{x}[f(x)-e^{x}]$ is a constant, then $f(x)$ is said to have property $M$ on $A$.  
$(1)$ Let $y=f(x)$ be an odd function with property $M$ on $R$. Find the analytical expression of $f(x)$.  
$(2)$ Let $y=g(x)$ be an even function with property $M$ on the interval $[-1,1]$. If the inequality $g(2x)-2eg(x)+n \gt 0$ holds for all $x \in [-1,1]$, find the range of real numbers for $n$.

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