
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
Three towns \(X, Y\), and \(Z\) lie on a plane with coordinates \((0,0)\), \((200,0)\), and \((0,300)\) respectively. There are 100, 200, and 300 students in towns \(X, Y\), and \(Z\) respectively. A school is to be built on a grid point \((x, y)\), where \(x\) and \(y\) are both integers, such that the overall distance travelled by all the students is minimized. Find the value of \(x + y\).

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