
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
We are looking for an important document in our desk. We know that \( p_{j} \) is the probability that the document is in the \( j \)-th drawer \(\left( j=1, 2, \ldots, n, \sum_{j=1}^{n} p_j = 1 \right) \) and that searching through the \( j \)-th drawer takes \( T_{j} \) time. Before starting the search, we create a plan by determining a permutation \( i_{1}, i_{2}, \ldots, i_{n} \) of the first \( n \) natural numbers and will search the drawers in this order until the document is found. With this plan, the expected search time is:

\[ 
K\left(i_{1}, i_{2}, \ldots, i_{n}\right)=\sum_{j=1}^{n} p_{i_{j}}\left(T_{i_{1}}+T_{i_{2}}+\ldots+T_{i_{j}}\right) 
\]

Determine the permutation \( i_{1}, i_{2}, \ldots, i_{n} \) for the given \( p_{1}, \ldots, p_{n}, T_{1}, \ldots, T_{n} \) values that minimizes \( K\left(i_{1}, i_{2}, \ldots, i_{n}\right) \).

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