
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 plane vectors $\overrightarrow{a}=(\sqrt{3},-1)$ and $\overrightarrow{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$. If there exist non-zero real numbers $k$ and $t$ such that $\overrightarrow{x}= \overrightarrow{a}+(t^{2}-3) \overrightarrow{b}$ and $\overrightarrow{y}=-k \overrightarrow{a}+t \overrightarrow{b}$, and $\overrightarrow{x} \perp \overrightarrow{y}$  
$(1)$ Find the function relationship $k=f(t)$;  
$(2)$ Find the range of values of $t$ for which $f(t) > 0$.

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