
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
The line $y=kx+1$ intersects the circle ${{\left( x-2 \right)}^{2}}+{{\left( y-1 \right)}^{2}}=4$ at points $P$ and $Q$. If the distance between $P$ and $Q$ is greater than or equal to $2\sqrt{2}$, determine the range of possible values for $k$.

A) $\left[ -\dfrac{3}{4},0 \right]$
B) $\left[ -\dfrac{\sqrt{3}}{3},\dfrac{\sqrt{3}}{3} \right]$
C) $\left[ -1,1 \right]$
D) $\left[ -\sqrt{3},\sqrt{3} \right]$

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