
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
It is known that the distance from any point on curve C to point P (2, 0) is 2 less than its distance to the line $x = -4$.
(Ⅰ) Find the equation of curve C;
(Ⅱ) A line passes through point P ($a$, 0) ($a > 0$) and intersects curve C at points A and B. Find the minimum value of the area of $\triangle AOB$.

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