
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
Suppose there are ten people, each carrying a bucket, who arrive at a water faucet to fill their buckets. Let \(T_i\) (for \(i=1,2, \ldots, 10\)) be the time required for the faucet to fill the bucket of the \(i\)-th person, with all of these \(T_i\) being distinct. How should these ten people be ordered when only one faucet is available so that their total spent time (including the time each individual spends fetching water) is minimized? What is this minimum time? Provide proof for your assertion.

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