
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 that an ellipse $C\_1$ with the center at the origin of the coordinate and a parabola $C\_2$: $x^2=4y$ share a focus $F\_1$, and a line $l$: $y=2x+m$ intersects the parabola $C\_2$ at only one point.
(1) Find the equation of the line $l$;
(2) If the ellipse $C\_1$ passes through a point $P$ on the line $l$, find the equation of the ellipse $C\_1$ when its eccentricity reaches the maximum.

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