
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
On a straight road, there are \(2n+1\) trees \(T_{1}, T_{2}, \cdots, T_{2n+1}\) (where \(n\) is a given positive integer) in sequence. A drunkard starts from the tree \(T_{n+1}\) in the middle and takes a random walk among these trees for \(n\) minutes according to the following rule: If at the end of a minute he is at tree \(T_{i} (2 \leqslant i \leqslant 2n)\), then at the end of the next minute he has probabilities \(\frac{1}{4}, \frac{1}{2}, \frac{1}{4}\) of being at positions \(T_{i-1}, T_{i}, T_{i+1}\) respectively.

(1) Find the probability that the drunkard is at each tree \(T_{i} (1 \leqslant i \leqslant 2n+1)\) at the end of the \(n\)-th minute.

(2) Assuming the distance between two adjacent trees is 1, find the expected distance between the drunkard's position at the end of the \(n\)-th minute and his initial position (i.e., tree \(T_{n+1}\)), expressed in the simplest form in terms of \(n\).

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