
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 vectors $\overrightarrow{m\_1}=(0,x)$, $\overrightarrow{n\_1}=(1,1)$, $\overrightarrow{m\_2}=(x,0)$, $\overrightarrow{n\_2}=(y^{2},1)$ (where $x$, $y$ are real numbers), and vectors $\overrightarrow{m}= \overrightarrow{m\_1}+ \sqrt {2} \overrightarrow{n\_2}$, $\overrightarrow{n}= \overrightarrow{m\_2}- \sqrt {2} \overrightarrow{n\_1}$, with $\overrightarrow{m}$ parallel to $\overrightarrow{n}$. The trajectory of the point $P(x,y)$ is a curve $C$.
(I) Find the equation of the curve $C$;
(II) Given line $l$: $y=kx+1$ intersects the curve $C$ at points $M$ and $N$, find the equation of the line $l$ when $|MN|= \frac {4 \sqrt {2}}{3}$.

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