
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
An equilateral triangle $ABC$ has another equilateral triangle $A_{1}B_{1}C_{1}$ inscribed in it. Point $A_{1}$ lies on side $BC$, point $B_{1}$ lies on side $AC$, and point $C_{1}$ lies on side $AB$. The angle $A_{1}B_{1}C$ is equal to $\alpha$. Find the ratio $AB$ to $A_{1}B_{1}$.

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