
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 \( E \) be a point moving inside the square \( ABCD \). Given that the minimum value of the sum of distances from \( E \) to the points \( A \), \( B \), and \( C \) is \( \sqrt{2} + \sqrt{6} \), find the side length of this square.

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