
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

If the spheres $\alpha, \beta, \gamma, \ldots$ passing through the circle $k$ intersect the lines $g$ and $g_{1}$, which pass through points $P$ and $P_{1}$ of the circle $k$, at points $A, B, C, \ldots$ and $A_{1}, B_{1}, C_{1} \ldots$ respectively, then

$$
\frac{A B}{A_{1} B_{1}}=\frac{B C}{B_{1} C_{1}}=\ldots=\text {constant.}
$$

If the lines $g$ and $g_{1}$ bend at equal angles to the plane of the circle $k$, then

$$
A B=A_{1} B_{1}, B C=B_{1} C_{1}, \ldots
$$

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