
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
Consider four points $A$, $B$, $C$, and $D$ such that triangles $ABC$, $BCD$, and $CDA$ are all equilateral triangles. If each side of the triangles measures $s$, find the ratio \( \frac{AD}{BC} \) when expressed in simplest radical form.

[asy]
draw((0,0)--(5,8.7)--(10,0)--cycle);
draw((10,0)--(15,8.7)--(5,8.7));
draw((15,8.7)--(0,0));
label("$A$",(0,0),SW);
label("$B$",(5,8.7),N);
label("$C$",(10,0),SE);
label("$D$",(15,8.7),NE);
[/asy]

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