
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
Let the function $f(x)=a^{x}-(k-1)a^{-x}$ ($a > 0$, $a \neq 1$) be an odd function defined on $\mathbb{R}$.

$(1)$ Find the value of $k$;

$(2)$ If $f(1) < 0$, determine the monotonicity of the function and find the range of values of $t$ for which the inequality $f(x^{2}+tx)+f(4-x) < 0$ always holds;

$(3)$ If $f(1)= \frac{3}{2}$, and $g(x)=a^{2x}+a^{-2x}-2mf(x)$, and $g(x)$ has a minimum value of $-2$ on $[1,+\infty)$, find the value of the real number $m$.

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