
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
Given the polar equation of curve $C_1$ is $\rho(\sqrt{2}\cos\theta - \sin\theta) = a$, and the parametric equation of curve $C_2$ is $\begin{cases} x=\sin\theta +\cos\theta \\ y=1+\sin2\theta \end{cases}$ (where $\theta$ is the parameter), and $C_1$ and $C_2$ have two different intersection points.  
(1) Write the rectangular coordinate equation of curve $C_1$ and the general equation of curve $C_2$;  
(2) Find the range of the real number $a$.

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