
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

On the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, different points $A\left(x_{1}, y_{1}\right), B\left(4, \frac{9}{5}\right), C\left(x_{2}, y_{2}\right)$ are such that their distances to the focus $F(4,0)$ form an arithmetic sequence. If the perpendicular bisector of segment $A C$ intersects the $x$-axis at point $T$, find the slope $k$ of the line $BT$. $\qquad$ .

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