
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 vectors $\overrightarrow{a} = (\sqrt{3}, -1)$ and $\overrightarrow{b} = (\frac{1}{2}, \frac{\sqrt{3}}{2})$, suppose 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 $\overrightarrow{x} \perp \overrightarrow{y}$. Find the minimum value of $\frac{k + t^2}{t}$.

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