
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 certain shipping company's ship spends on fuel per hour in direct proportion to the square of the ship's speed, with a proportionality constant of $k$. The maximum speed of the ship is 15 nautical miles per hour. When the ship's speed is 10 nautical miles per hour, its fuel cost is $96 per hour, and the rest of the operating costs (regardless of speed) total $150 per hour. Assume that the ship sails at a constant speed $v$ during operation.  
(1) Find the value of $k$;  
(2) Find the minimum total cost $W$ (fuel cost + operating cost) for the ship to travel 100 nautical miles.

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