
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
Given an odd function \( f(x) \) defined on \( (-\infty, 0) \cup (0, +\infty) \), which is decreasing on the interval \( (0, +\infty) \), and satisfies \( f(k) = 0 \) for \( k > 0 \). Additionally, consider the function \( \varphi(\alpha) = \sin^{2} \alpha + (\xi + 1) \cos \alpha - \xi^{2} - \xi - k \) for \( \alpha \in [0, \pi] \). Let \( A = \{ \xi \mid \varphi(\alpha) < 0 \text{ for all } \alpha \in [0, \pi] \} \) and \( B = \{ \xi \mid f(\varphi(\alpha)) > 0 \text{ for all } \alpha \in [0, \pi] \} \). Find \( A \cap B \).

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