
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 vectors $\overrightarrow{AB} = (1, 2\cos\theta)$ and $\overrightarrow{BC} = (m, -4)$, where $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, and for any $m \in [-1, 0]$, the dot product $\overrightarrow{AC} \cdot \overrightarrow{BC} \leq 10$ holds true. Find the maximum value of $\sin(\theta - \frac{\pi}{2})$.

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