
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
Let \( A_{1}, A_{2}, \ldots, A_{2015} \) be distinct points on the unit circle with center \( O \). For every two distinct integers \( i, j \), let \( P_{ij} \) be the midpoint of \( A_{i} \) and \( A_{j} \). Find the smallest possible value of
$$
\sum_{1 \leq i < j \leq 2015} O P_{ij}^{2}.
$$

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