
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 $P$ be a moving point on the plane, and denote its trajectory as $\Gamma$. <br/>$(1)$ Please choose one of the following three conditions and find the corresponding equation of $\Gamma$; <br/>① A moving circle with center at point $P$ passes through point $F(-1,0)$ and is tangent to the circle $K: (x-1)^2 + y^2 = 16$; <br/>② Given point $T(-1,0)$, line $l: x=-4$, the ratio of the distance from moving point $P$ to point $T$ to the distance from $P$ to line $l$ is $\frac{1}{2}$; <br/>③ Let $E$ be a moving point on the circle $O: x^2 + y^2 = 4$, a line $EG$ is drawn through $E$ perpendicular to the $x$-axis, with foot $G$, and $\overrightarrow{GP} = \frac{\sqrt{3}}{2}\overrightarrow{CE}$. <br/>$(2)$ Under condition (1), let the left and right vertices of the curve $\Gamma$ be $A$ and $B$ respectively. If a line $m$ passing through point $K(1,0)$ has a non-zero slope, and line $m$ intersects curve $\Gamma$ at points $M$ and $N$, a line $n$ passes through point $T(-1,0)$ and is perpendicular to the $x$-axis. Line $AM$ intersects line $n$ at point $P$, and line $BN$ intersects line $n$ at point $Q$. Is the ratio $\frac{|TP|}{|TQ|}$ a constant value? If yes, find the constant value; if not, explain why.

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