
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 rectangular coordinate system on a plane, curve $C\_1$: $\begin{cases} x=2t+2a \\ y=-t \end{cases} (t \text{ is a parameter} )$, and curve $C\_2$: $\begin{cases} x=2\sin \theta \\ y=1+2\cos \theta \end{cases} (\theta \text{ is a parameter} )$. If curves $C\_1$ and $C\_2$ have a common point, then the range of real number $a$ is _______.

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