
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 $f(x)=2\sin(\omega x+\phi)$ $(\omega > 0, 0 < \phi < \frac{\pi}{2})$, $f(x_1)=2$, $f(x_2)=0$, if the minimum value of $|x_1-x_2|$ is $\frac{1}{2}$, and $f(\frac{1}{2})=1$, determine the interval in which $f(x)$ is monotonically increasing.

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