
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]


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In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C\_1$ are given by $\begin{cases} x=t\cos \alpha \ y=-1+t\sin \alpha \end{cases}$ where $t$ is a parameter and $0\leqslant \alpha < \pi$. In the polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of the curve $C\_2$ is given by $\rho =2\sqrt{2}\cos (\theta -\dfrac{\pi }{4})$.

(I) Find the Cartesian equation of $C\_2$ and describe the shape of its graph.
(II) The curves $C\_1$ and $C\_2$ intersect at two distinct points $A$ and $B$. The midpoint of the line segment $AB$ is $M$, and there is a point $N(0,{-}1)$. If $|MN|=2$, find the value of $\sin\alpha$ in the parametric equations of $C\_1$.

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