
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the focus $F$ of the parabola $M: x^2 = 4y$. $F_1$ and $F_2$ are the left and right foci of the ellipse $C$, respectively, and $\overrightarrow{F_1F} \cdot \overrightarrow{F_1F_2} = 6$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ If a line $l$ is tangent to the parabola $M$ and intersects the ellipse $C$ at points $A$ and $B$, find the maximum value of the area of $\triangle OAB$.

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