
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 $f(x)=\cos(\omega x+\varphi)$ where $(\omega>0,0 < \varphi < \pi)$ has a minimum positive period of $T$, if $f(T)=\frac{1}{2}$ and $x=\frac{7\pi}{3}$ is a critical point of the function $f(x)$, calculate the minimum value of $\omega$.

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