
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
What is the largest negative \( a \) for which the inequality \(\frac{\sqrt[3]{\cos x} - \sqrt[3]{\sin x}}{\sqrt[3]{\operatorname{tg} x} - \sqrt[3]{\operatorname{tg} x}} > a\) is satisfied for all permissible \( x \in \left(-3 \pi, -\frac{5 \pi}{2}\right) \)? Round the answer to the nearest hundredth if necessary.

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