
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 $\sin\left(\frac{\pi}{3} + \frac{\alpha}{6}\right) = -\frac{3}{5}$ and $\cos\left(\frac{\pi}{6} + \frac{\beta}{2}\right) = -\frac{12}{13}$, with $-5\pi < \alpha < -2\pi$ and $-\frac{\pi}{3} < \beta < \frac{5\pi}{3}$, find the value of $\sin\left(\frac{\alpha}{6} + \frac{\beta}{2}\right)$.

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