
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
A sledding hill consists of a straight incline \( AB \) and a horizontal section \( BC \). Point \( A \) is located 5 meters from the nearest horizontal ground point \( H \). The distance \( HC \) is 3 meters, with point \( B \) lying on the segment \( HC \). Find the distance from point \( H \) to point \( B \) such that the time it takes for the sled to move from rest along the broken line \( ABC \) is minimized. Assume a uniform gravitational field, and neglect friction, air resistance, and changes in the sled's speed vector as it passes through point \( B \). Assume the acceleration due to gravity is \( 10 \ \text{m/s}^2 \).

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