
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 shortest distance from a point on curve $C_1$ with parametric equation 
$$
\begin{cases} 
x = 1 + \cos\theta \\ 
y = \sin\theta 
\end{cases} (\theta \text{ is the parameter})
$$
to a point on curve $C_2$ with parametric equation 
$$
\begin{cases} 
x = -2\sqrt{2} + \frac{1}{2}t \\ 
y = 1 - \frac{1}{2}t 
\end{cases} (t \text{ is the parameter})
$$
is ______.

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