
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 $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \dfrac{\sqrt{2}}{2}$, the length of the minor axis is $2$.  
$(1)$ Find the equation of the ellipse;  

$(2)$ Point $A$ is a moving point on the ellipse (not the endpoints of the major axis), the extension line of $AF_2$ intersects the ellipse at point $B$, and the extension line of $AO$ intersects the ellipse at point $C$. Find the maximum value of the area of $\triangle ABC$.

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