
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 a plane, triangles $ABC$ and $BCD$ are arranged such that $ABC$ is an equilateral triangle and $BCD$ is an isosceles triangle with $BC = BD$. If $AB = BC = a$, what is the value of $AD \div BC$ when expressed in simplest radical form?

[asy]
pair A, B, C, D;
A = (0,0);
B = (5,8.7);
C = (10,0);
D = (20,0);
draw(A--B--C--D--B);
label("$A$", A, SW);
label("$B$", B, N);
label("$C$", C, SE);
label("$D$", D, SE);
[/asy]

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