
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 the parabola $C_1: y^2 = 4px$ ($p > 0$) with focus $F_2$, and its directrix intersects the x-axis at point $F_1$. Let the ellipse $C_2$ have $F_1$ and $F_2$ as its left and right foci, respectively, with an eccentricity $e = \frac{1}{2}$. Moreover, the parabola $C_1$ and ellipse $C_2$ intersect at point $M$.
(1) When $p=1$, find the standard equation of the ellipse $C_2$;
(2) Under the condition of (1), if the line $l$ passes through the right focus $F_2$ of ellipse $C_2$ and intersects the parabola $C_1$ at points $A$ and $B$, and if the chord length $|AB|$ equals the perimeter of triangle $\triangle MF_1F_2$, find the equation of line $l$.

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