
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
It is known that $F_1$ and $F_2$ are the upper and lower foci of the ellipse $C: \frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$ ($a>b>0$), 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 $A(b,0)$, $B(0,a)$, 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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