
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
There are ten people, each with a bucket, who need to fill their buckets from a water tap. The time required to fill the bucket for the $i^{th}$ person $(i=1,2, \cdots, 10)$ is $T_{i}$ minutes, and all $T_{i}$ values are distinct. 

(1) When only one water tap is available, how should the ten people be arranged in sequence to minimize the total time spent (including each individual's time to fill their own bucket)? What is this minimum total time? (Proof is required for your conclusion.)

(2) When two water taps are available, how should the ten people be arranged in sequence to minimize the total time spent? What is this minimum total time? (Proof is required for your conclusion.)

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