
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
Let $k\in \mathbb{N}$, $f(x)=\sin(\frac{2k+1}{3}\pi x)$. If for any real number $a$, the function $y=f\left(x\right)$ has the value $\frac{1}{3}$ at least $2$ times and at most $6$ times in the interval $\left[a,a+3\right]$, then determine the possible values of $k$.

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