
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 that the terminal side of angle $\varphi$ passes through point P(1, -1), and points A($x_1$, $y_1$) and B($x_2$, $y_2$) are any two points on the graph of the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$). If $|f(x_1) - f(x_2)| = 2$, the minimum value of $|x_1 - x_2|$ is $\frac{\pi}{3}$. Find the value of $f\left(\frac{\pi}{2}\right)$.

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