
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 vectors $\overrightarrow{a}=( \frac { \sqrt {3}}{2},- \frac {1}{2})$ and $\overrightarrow{b}=(1, \sqrt {3})$, with $\overrightarrow{x}= \overrightarrow{a}+(t^{2}-3) \overrightarrow{b}$ and $\overrightarrow{y}=-k \overrightarrow{a}+t \overrightarrow{b}$, find an expression for $k$ in terms of $t$ if $\overrightarrow{x}$ and $\overrightarrow{y}$ are perpendicular.

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