
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 $f(x) = 2\sin(\omega x + \frac{\pi}{6})$ ($\omega > 0$), its graph intersects the y-axis at point P and intersects the x-axis at two adjacent points, denoted as A and B. If the area of $\triangle PAB$ equals $\pi$, then $\omega = \_\_\_\_\_\_$.

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