
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, a fixed point $A(1,0)$ and a moving point $P$ are given. The circle with $PA$ as diameter is always tangent to the $y$-axis. Let the trajectory of the moving point $P$ be curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Point $Q(x_0,y_0) (x_0 \geqslant 5)$ is a point on curve $C$. Through point $Q$, two tangent lines to the circle $E: (x-2)^2+y^2=4$ are drawn, intersecting the $x$-axis at points $M$ and $N$, respectively. Find the minimum value of the area of $\triangle QMN$.

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