
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
Two villages, \( A \) and \( B \), lie on opposite sides of a straight river. Village \( B \) will host a market, and residents of village \( A \) wish to attend. The residents of \( A \) would like to build a bridge perpendicular to the river so that the total route walked from \( A \) to the bridge, across the bridge, and onward to \( B \) is as short as possible. How can they find the exact location of the bridge, and how long will the route be?

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