
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 small villages, \(A\) and \(B\), are located along a straight 10 km section of a railway track. Despite the flat terrain, the road connecting the villages is quite winding. A cyclist noticed that at any point \(P\) on this road, the sum of the length \(BH\) (multiplied by 20), where \(H\) is the projection of the cyclist's position \(P\) on the railway track, the cube of the same length (multiplied by 13), and the length \(PH\) (multiplied by 300) is equal to the sum of the length \(PH\) raised to the fourth power and the square of this length (multiplied by 32) (all lengths are in kilometers).

How many railway crossings does the cyclist encounter on the route from \(A\) to \(B\)?

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