
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 following three conditions:① The function $f(x-\frac{π}{3})$ is an odd function; ② When $x=\frac{π}{6}$, $f\left(x\right)=2$; ③ $\frac{2π}{3}$ is a zero of the function $f\left(x\right)$. Choose one of these three conditions to fill in the blank below and answer the following questions.<br/>Given the function $f(x)=2\sin(x+φ)(0＜φ＜\frac{π}{2})$, _____.<br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$;<br/>$(2)$ Find the intervals where the function $f\left(x\right)$ is monotonically increasing on $\left[0,2\pi \right]$.

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