
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 $f(x)=\frac{2x}{x^2+6}$.

$(1)$ If the solution set of $f(x) > k$ is $\{x|x < -3$ or $x > -2\}$, find the value of $k$;

$(2)$ For any $x > 0$, $f(x) \leqslant t$ always holds, find the range of values for $t$.

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