
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 $f\left(x\right)=\sin \left(\omega x+\varphi \right)$ is monotonically increasing on the interval $\left(\frac{π}{6},\frac{2π}{3}\right)$, and the lines $x=\frac{π}{6}$ and $x=\frac{2π}{3}$ are two symmetric axes of the graph of the function $y=f\left(x\right)$, find $f\left(0\right)$.

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