
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 $f(x)=\frac{{2{e^x}}}{{{e^x}+1}}+k$ is an odd function. $(e$ is the base of the natural logarithm).<br/>$(1)$ Find the value of the real number $k$;<br/>$(2)$ If $x \gt 0$, the inequality $f\left(2x\right)\leqslant mf\left(x\right)$ always holds for $x$. Find the range of the real number $m$;<br/>$(3)$ Let $g(x)=\frac{{f(x)+1}}{{1-f(x)}}$. For any real numbers $a$, $b$, $c\in \left(0,n\right]$, if segments of lengths $a$, $b$, $c$ can form a triangle, then segments of lengths $g\left(a\right)$, $g\left(b\right)$, $g\left(c\right)$ can also form a triangle. Find the maximum value of the real number $n$.

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