
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 \( a = \sin 1^\circ \cdot \sin 2^\circ \cdot \sin 3^\circ \cdot \ldots \cdot \sin 89^\circ \) and \( b = \sin 1^\circ \cdot \sin 3^\circ \cdot \ldots \cdot \sin 87^\circ \cdot \sin 89^\circ \). Then, \( \frac{a}{b} = \sin 2^\circ \cdot \sin 4^\circ \cdot \sin 6^\circ \cdot \ldots \cdot \sin 88^\circ \).

Applying the double angle formula, we find:

\[ 
\frac{a}{b} = 2^{44} \cdot \sin 1^\circ \cdot \sin 2^\circ \cdot \sin 3^\circ \cdot \ldots \cdot \sin 44^\circ \cdot \cos 1^\circ \cdot \cos 2^\circ \cdot \cos 3^\circ \cdot \ldots \cdot \cos 44^\circ
\]

From which it follows:

\[
\begin{aligned}
& b = \sin 1^\circ \cdot \sin 3^\circ \cdot \ldots \cdot \sin 87^\circ \cdot \sin 89^\circ = \\
& = \frac{a}{2^{44} \cdot \sin 1^\circ \cdot \cos 1^\circ \cdot \sin 2^\circ \cdot \cos 2^\circ \ldots \cdot \sin 44^\circ \cdot \cos 44^\circ} = \\
& = \frac{\sin 1^\circ \cdot \sin 2^\circ \cdot \ldots \cdot \sin 89^\circ}{2^{44} \cdot \sin 1^\circ \cdot \cos 1^\circ \cdot \sin 2^\circ \cdot \cos 2^\circ \cdot \ldots \cdot \sin 44^\circ \cdot \cos 44^\circ} = \\
& = \frac{\sin 45^\circ \cdot \sin 46^\circ \cdot \ldots \cdot \sin 89^\circ}{2^{44} \cdot \cos 1^\circ \cdot \cos 2^\circ \cdot \ldots \cdot \cos 44^\circ} = \\
& = \frac{\sin 45^\circ \cdot \sin 46^\circ \cdot \ldots \cdot \sin 89^\circ}{2^{44} \cdot \sin 89^\circ \cdot \sin 88^\circ \cdot \ldots \cdot \sin 46^\circ} = \frac{\frac{\sqrt{2}}{2}}{2^{44}} = \frac{\sqrt{2}}{2^{45}}
\end{aligned}
\]

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