
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
Let \( ABCD \) be a convex quadrilateral and \( Z \) a point inside it. Denote by \( \alpha_1 \) the angle \( \widehat{ZAD} \), \( \alpha_2 \) the angle \( \widehat{ZAB} \), \( \beta_1 \) the angle \( \widehat{ZBA} \), \( \beta_2 \) the angle \( \widehat{ZBC} \), \( \gamma_1 \) the angle \( \widehat{ZCB} \), \( \gamma_2 \) the angle \( \widehat{ZCD} \), \( \delta_1 \) the angle \( \widehat{ZDC} \), and \( \delta_2 \) the angle \( \widehat{ZDA} \). Show the following equality:

$$
\frac{\sin \alpha_1}{\sin \alpha_2} \cdot \frac{\sin \beta_1}{\sin \beta_2} \cdot \frac{\sin \gamma_1}{\sin \gamma_2} \cdot \frac{\sin \delta_1}{\sin \delta_2} = 1
$$

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