
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
For the curve $C_1$ with the parametric equations \(\left\{\begin{array}{l}
x=1+\cos\theta \\
y=\sin\theta
\end{array}\right.\) (where \(\theta\) is the parameter), find a point on curve $C_1$ such that its distance to the line $C_2$ with the parametric equations \(\left\{\begin{array}{l}
x=-2\sqrt{2}+\frac{1}{2}t \\
y=1-\frac{1}{2}t
\end{array}\right.\) (where \(t\) is the parameter) is minimized, and then determine the coordinates of this point and the minimum distance.

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