
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
In triangle \( \triangle ABC \), \( AB = AC = 2 \). On the side \( BC \), there are 100 distinct points \( P_1, P_2, \cdots, P_{100} \). Define \( m_i = AP_i^2 + BP_i \cdot P_iC \) for \( i = 1, 2, \cdots, 100 \). Find the value of \( m_1 + m_2 + \cdots + m_{100} \).

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