
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
Given a population of $n$ elements where $n \geq 4$, denoted as $\{1, 2, 3, \ldots, n\}$, perform a sampling by dividing the population into two subpopulations $\{1, 2, 3, \ldots, m\}$ and $\{m+1, m+2, \ldots, n\}$ (where $m$ is a given positive integer, and $2 \leq m \leq n-2$), and then randomly select 2 elements from each subpopulation to form a sample. Let $P_{ij}$ represent the probability that elements $i$ and $j$ appear together in the sample.
1. Find the expression for $P_{1n}$ in terms of $m$ and $n$.
2. Calculate the sum of all $P_{ij}$ for $1 \leq i < j \leq n$.

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