
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
In the Cartesian coordinate system $xOy$, if there exists a point $P$ on the line $y=k(x-3\sqrt{3})$ and a point $Q$ on the circle $x^{2}+(y-1)^{2}=1$ such that $\overrightarrow{OP}=3\overrightarrow{OQ}$, then the minimum value of the real number $k$ is $\boxed{-\sqrt{3}}$.

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