
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 first tourist's motion law is described by \( S = \sqrt{1 + 6t} - 1 \), and the second tourist's motion law 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 if and only if one tourist is on one side of a road sign and the other tourist is on the other side (either could also be exactly at the sign).

Since the first tourist reaches the sign at time \( t_{1} \) such that \( \sqrt{1 + 6t_{1}} - 1 = 2 \), which is at \( t_{1} = \frac{4}{3} \), and the second tourist reaches the sign at time \( t_{2} \) such that \( 6\left(t_{2} - \frac{1}{6}\right) = 2 \), which is at \( t_{2} = \frac{1}{2} \), the suitable time intervals are \( t \in \left[\frac{1}{2}, \frac{4}{3}\right] \), i.e., from half an hour to 1 hour 20 minutes after the first tourist starts.

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