
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 a plane Cartesian coordinate system, consider curve $C_1$ represented by the parametric equations:
$$
C_1: \begin{cases}
x = 2t + 2a \\
y = -t
\end{cases}
(t \text{ is the parameter}),
$$
and curve $C_2$ represented by the parametric equations:
$$
C_2: \begin{cases}
x = 2\cos\theta \\
y = 2 + 2\sin\theta
\end{cases}
(a \text{ is the parameter}).
$$

If the curves $C_1$ and $C_2$ have a common point, find the range of values for the real number $a$.

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