
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 the function \( f(x)=\cos x + m \left(x+\frac{\pi}{2}\right) \sin x \) where \( m \leqslant 1 \):

(1) Discuss the number of zeros of \( f(x) \) in the interval \( (-\pi, 0) \).

(2) If there exists \( t > 0 \) such that \( |f(x)| < -2x - \pi \) holds for \( x \in \left(-\frac{\pi}{2} - t, -\frac{\pi}{2}\right) \), determine the minimum value of \( m \) that satisfies this condition.


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