
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
For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary.

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