
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 \( \mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{41} \) be 41 distinct points on the segment \( \mathrm{BC} \) of a triangle \( \mathrm{ABC} \), where \( \mathrm{AB} = \mathrm{AC} = 7 \). Evaluate the sum \( \sum_{i=1}^{41}\left(\mathrm{AP}_{i}^{2}+\mathrm{P}_{i} \mathrm{~B} \cdot \mathrm{P}_{i} \mathrm{C}\right) \).

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