
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 a moving point $P(x,y)$, where $x \geq 0$, be such that its distance to the fixed point $F\left(\frac{1}{2}, 0\right)$ is greater by $\frac{1}{2}$ than its distance to the y-axis. Let the trajectory of point P be the curve C.
(Ⅰ) Find the equation of the trajectory of point P;
(Ⅱ) Let circle M pass through the point $A(1,0)$, with the center M on the trajectory of P. Let BD be the chord intercepted by circle M on the y-axis. Is the length of chord BD constant as M moves? Explain your reasoning;
(Ⅲ) Draw two perpendicular lines through $F\left(\frac{1}{2}, 0\right)$ that intersect curve C at points G, H, R, and S. Find the minimum area of quadrilateral GRHS.

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