
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
Elective question: Given the curve $C_1$: 
$$
\begin{cases} 
x=\cos\theta \\ 
y=\sin\theta
\end{cases}
$$
($\theta$ is the parameter), and the curve $C_2$: 
$$
\begin{cases} 
x= \frac { \sqrt {2}}{2}t- \sqrt {2} \\ 
y= \frac { \sqrt {2}}{2}t
\end{cases}
$$
($t$ is the parameter).

(I) Identify what curves $C_1$ and $C_2$ are, and state the number of common points between $C_1$ and $C_2$;

(II) If the y-coordinates of the points on $C_1$ and $C_2$ are compressed to half of their original values, resulting in curves $C_1'$ and $C_2'$ respectively. Write down the parametric equations for $C_1'$ and $C_2'$. Is the number of common points between $C_1'$ and $C_2'$ the same as the number of common points between $C_1$ and $C_2$? Explain your reasoning.

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