
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 polar coordinates, the "polar coordinate midpoint" of two points \\((ρ\_{1},θ\_{1})\\) and \\((ρ\_{2},θ\_{2})(ρ\_{1},ρ\_{2} > 0,0\\leqslant θ\_{1},θ\_{2}\\leqslant 2π)\\) is defined as \\(( \\dfrac {ρ\_{1}+ρ\_{2}}{2}, \\dfrac {θ\_{1}+θ\_{2}}{2})\\). Let points A and B have polar coordinates \\((4, \\dfrac {π}{100})\\) and \\((8, \\dfrac {51π}{100})\\), respectively. Let M be the midpoint of segment AB, and N be the "polar coordinate midpoint" of points A and B. Calculate the square of the length of segment MN.

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