
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 points A, B, C, and D on a spherical surface with a radius of 2, such that AB is perpendicular to AC, AD is perpendicular to AC, and AB is perpendicular to AD. Determine the maximum value of the sum of the areas of the three triangles S$_{m △ABC}$ + S$_{m △ABD}$ + S$_{m △ACD}$.

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