
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

The law of motion for the first tourist is given by $S=\sqrt{1+6 t}-1$, and for the second tourist, it is $-S=6\left(t-\frac{1}{6}\right)$ for $t \geq \frac{1}{6}$; $S=0$ for $t<\frac{1}{6}$.

The required condition is met when both tourists are on the same side of the sign, and not met when the sign is between them. The first tourist reaches the sign at time $t_{1}$ where $\sqrt{1+6 t_{1}}-1=2$, resulting in $t_{1}=\frac{4}{3}$. The second tourist reaches the sign at time $t_{2}$ where $6\left(t_{2}-\frac{1}{6}\right)=2$, resulting in $t_{2}=\frac{1}{2}$. Therefore, the suitable times are $t \in\left[0, \frac{1}{2}\right] \cup\left[\frac{4}{3},+\infty\right]$.

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