
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
Define ${min}{ a,b }=\begin{cases} a,& \left( a\leqslant b \right) \\ b,& (a > b) \\ \end{cases}$, if the function $f\left( x \right)={min}{ {\sin } \left( 2x+\frac{\pi }{6} \right),{\cos }2x }$, and the range of $f\left( x \right)$ on the interval $\left[ s,t \right]$ is $\left[ -1,\frac{1}{2} \right]$, calculate the maximum length of the interval $\left[ s,t \right]$.

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