
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 polar coordinate system, the polar equation of curve $C_1$ is $\rho \sin(\theta - \frac{\pi}{4}) = \sqrt{2}$. If we establish a Cartesian coordinate system with the pole at the origin and the polar axis along the x-axis, then the Cartesian equation of $C_1$ is _______________; the parametric equations of curve $C_2$ in the Cartesian coordinates are $\begin{cases} x = 2\cos t \\ y = 2 + 2\sin t \end{cases}$ (parameter $t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$), then the Cartesian equation of $C_2$ is _______________; the length of the chord cut by $C_2$ from $C_1$ is _____.

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