
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 ellipse $E$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ intersects with the positive half of the $y$-axis at point $M$, and points $F_1$, $F_2$ are the foci of the ellipse, and $\triangle MF_1F_2$ is an equilateral triangle with side length $2$. If the line $l$: $y=kx+2\sqrt{3}$ intersects the ellipse $E$ at two distinct points $A$, $B$.
$(1)$ Is the product of the slopes of lines $MA$, $MB$ a constant value? If yes, find this constant value. If no, explain why.
$(2)$ Find the maximum area of $\triangle ABM$.

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