
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
In the Cartesian coordinate system $xOy$, two points $E(-1, 0)$ and $F(1, 0)$ are given. A moving point $M$ satisfies $\overrightarrow{EM} \cdot \overrightarrow{FM} = 0$. Let the trajectory of point $M$ be $C$, and the semi-parabola $C'$: $y^2 = 2x$ ($y \geq 0$). Let point $D\left( \frac{1}{2}, 0\right)$.  
(Ⅰ) Find the equation of the trajectory $C$;  
(Ⅱ) Let point $T$ be a point on the curve $C'$, the tangent line to $C'$ at point $T$ intersects curve $C$ at points $A$ and $B$. Find the maximum value of the area of $\triangle ABD$ and the coordinates of point $T$.

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