
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 an ellipse $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b>0$) with its left and right foci as $F_1$ and $F_2$ respectively, where $F_2$ is also the focus of the parabola $C_2: y^2 = 4x$. Let $M$ be the intersection point of $C_1$ and $C_2$ in the first quadrant, and $|MF_2| = 2$.  
(1) Find the equation of the ellipse $C_1$.  
(2) Given that the vertices $A$ and $C$ of the rhombus $ABCD$ are on the ellipse $C_1$, and the vertices $B$ and $D$ are on the line $7x - 7y + 1 = 0$, find the equation of the line $AC$.

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