
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 $xoy$, the ratio of the distance between a moving point $P$ and a fixed point $F(1,0)$ to its distance from a fixed line $x=2$ is $\frac{\sqrt{2}}{2}$.
(Ⅰ) Find the equation of the trajectory $\Gamma$ of the moving point $P$;
(Ⅱ) Suppose on the curve $\Gamma$, three points $A(x_1,y_1)$, $B(1,\frac{\sqrt{2}}{2})$, and $C(x_2,y_2)$ have distances to point $F$ that form an arithmetic sequence. The perpendicular bisector of segment $AC$ intersects the $x$-axis at point $T$. Find the slope $k$ of line $BT$.

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