
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 square with sides of length 4, calculate the sum of distances from one vertex to each of the midpoints of the sides and to both vertices diagonally opposite this vertex.
**A)** \( 10 + 4\sqrt{5} \)  
**B)** \( 12 + 4\sqrt{5} \)  
**C)** \( 14 + 4\sqrt{5} + 4\sqrt{2} \)  
**D)** \( 10 + 4\sqrt{5} + 4\sqrt{2} \)  
**E)** \( 8 + 4\sqrt{5} + 4\sqrt{2} \)

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