
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
Due to the increasing personnel exchanges between Hefei and Nanjing as a result of urban development, there is a plan to build a dedicated railway to alleviate traffic pressure, using a train as a shuttle service. It is known that the daily round-trip frequency $y$ of the train is a linear function of the number of carriages $x$ it tows each time. If 4 carriages are towed, the train runs 16 times a day, and if 7 carriages are towed, the train runs 10 times a day.
Ⅰ. Find the functional relationship between the daily round-trip frequency $y$ and the number of carriages $x$;
Ⅱ. Find the function equation for the total number $S$ of carriages operated daily with respect to the number of carriages $x$ towed;
Ⅲ. If each carriage carries 110 passengers, how many carriages should be towed to maximize the number of passengers transported daily? Also, calculate the maximum number of passengers transported daily.

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