
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 $F\_1$ and $F\_2$ are the upper and lower foci of the ellipse $C$: $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 (a > b > 0)$, respectively, where $F\_1$ is also the focus of the parabola $C\_1$: $x^2 = 4y$. Point $M$ is the intersection of $C\_1$ and $C\_2$ in the second quadrant, and $|MF\_1| = \frac{5}{3}$.
1. Find the equation of the ellipse $C\_1$;
2. Given $A(b, 0)$ and $B(0, a)$, the line $y = kx (k > 0)$ intersects $AB$ at point $D$, and intersects the ellipse $C\_1$ at points $E$ and $F$. Find the maximum area of the quadrilateral $AEBF$.

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