
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 rectangular coordinate system $(xOy)$, the left focus of the ellipse $C\_1: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\,(a > b > 0)$ is $F\_1(-1,0)$, and point $P(0,1)$ is on $C\_1$.

(I) Find the equation of ellipse $C\_1$;

(II) If line $l$ is tangent to both ellipse $C\_1$ and parabola $C\_2$: $y^{2}=4x$, find the equation of line $l$.

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