
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 \(\cos 2 \alpha\), given that \(2 \operatorname{ctg}^{2} \alpha+7 \operatorname{ctg} \alpha+3=0\) and the value of \(\alpha\) satisfies the inequalities: 
a) \(\frac{3 \pi}{2}<\alpha<\frac{7 \pi}{4}\);
b) \(\frac{7 \pi}{4}<\alpha<2 \pi\).

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