
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 $f(x) = 2\sin(\omega x + \phi) + 1$ with $\omega > 0$ and $|\phi| \leq \frac{\pi}{2}$, the shortest distance between the two nearest intersection points of its graph and the line $y=2$ is $\frac{\pi}{3}$. If $f(x) > 1$ holds for all $x \in \left(-\frac{\pi}{8}, \frac{\pi}{3}\right)$, determine the range of $\phi$.

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