
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
Given $ABCD$ be a kite with $AB=AD$ and $BC=CD$, with diagonals intersecting at point $F$. Let side $AD$ be extended through $D$, and side $BC$ be extended through $C$, both meeting at point $E$. Define angle sum $S$ as the degree sum of angles $CDE$ and $DCE$, and angle sum $S'$ as the sum of angles $BAD$ and $ABC$. Given that $\angle BDC = \theta$, determine the value of the ratio $r = S / S'$.

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