
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 a new scenario, Veronica is seeking to verify the coordinates of the Fermat point \(Q\) of triangle \(\triangle DEF\), where point \(D\) is at the origin, \(E\) is at \((6, 4)\), and \(F\) is at \((3, -2)\). She hypothesizes that point \(Q\) at \((2,1)\) minimizes the total distance. Compute the sum of distances from \(Q\) to the vertices of \(\triangle DEF\). If the total distance can be expressed as \(a + b\sqrt{c}\), where \(a\) and \(b\) are integers and \(c\) a minimized integer, find \(a + b\).

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