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

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

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

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