
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
Let the parabola $C: y^2 = 2px$ ($p>0$) have the focus $F$, the directrix $l$, and $M$ lie on $C$. The circle with center $M$ is tangent to the directrix $l$ at point $Q$, whose ordinate is $\sqrt{3}p$. Point $E(5,0)$ is another intersection of the circle with the x-axis, other than point $F$.
(Ⅰ) Find the equations for parabola $C$ and circle $M$.
(Ⅱ) Line $n$ passes through $F$ with a slope of $\frac{4}{3}$ and intersects $C$ at two points $A$ and $B$. Find the area of $\triangle ABQ$.

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