
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
Find the value of the following expressions:
(1) $\sin 10^{\circ} \cdot \sin 30^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ}$;
(2) $\sin ^{2} 20^{\circ}+\cos ^{2} 80^{\circ}+\sqrt{3} \sin 20^{\circ} \cdot \cos 80^{\circ}$;
(3) $\cos ^{2} A+\cos ^{2}\left(60^{\circ}-A\right)+\cos ^{2}\left(60^{\circ}+A\right)$;
(4) $\cos \frac{\pi}{15} \cdot \cos \frac{2 \pi}{15} \cdot \cos \frac{3 \pi}{15} \cdot \cos \frac{4 \pi}{15} \cdot \cos \frac{5 \pi}{15} \cdot \cos \frac{6 \pi}{15} \cdot \cos \frac{7 \pi}{15}$.

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