
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 Cartesian coordinate system $(xOy)$, the parametric equations of curve $C$ are given by $\begin{cases} x=\sin\alpha+\cos\alpha \ y=\sin\alpha-\cos\alpha \end{cases}$ $(\alpha$ is the parameter$)$.

1. Find the Cartesian equation of curve $C$.
2. In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the equation of line $l$ is given by $\sqrt{2}\rho\sin(\frac{\pi}{4}-\theta)+\frac{1}{2}=0$. It is known that line $l$ intersects curve $C$ at points $A$ and $B$. Find $|AB|$.

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