
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 minimum value of the expression

$$
A=\frac{\operatorname{ctg} 2 \alpha - \operatorname{tg} 2 \alpha}{1+\sin \left(\frac{5 \pi}{2} - 8 \alpha\right)}
$$

for \( 0 < \alpha < \frac{\pi}{8} \.

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