
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 $C:\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1 \left( a > b > 0 \right)$ shares a focus with the parabola $y^2=4\sqrt{6}x$, and a chord passing through the right focus $F$ of the ellipse and perpendicular to the x-axis has a length of $2$.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) If a line $l$ perpendicular to the line $l_1:x-2y+t=0$ intersects the ellipse $C$ at points $B$ and $D$, find the maximum value of $\Delta OBD$.

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