
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
The exponential function $y=g(x)$ satisfies $g(2)=4$, and the function $f(x)= \frac{-g(x)+n}{2g(x)+m}$, whose domain is $\mathbb{R}$, is an odd function.  
$(1)$ Find the values of the real numbers $m$ and $n$;  
$(2)$ If there exists a real number $t$, such that the inequality $f(t^{2}-2t)+f(2t^{2}-k) > 0$ holds, find the range of the real number $k$.

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