
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 $\omega \in N$, $\varphi \in [0,\pi]$, there are exactly two extreme points in the interval $x \in \left(-\frac{2}{3}\pi,\frac{1}{3}\pi\right)$, and $f\left(-\frac{2}{3}\pi\right)+f\left(\frac{1}{3}\pi\right)=0$. Find the set of all possible values of $\varphi$.

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