
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 circles, each having a radius of $2$, are placed with centers at $(10, 50)$, $(12, 34)$, and $(15, 40)$. A line passing through $(12,34)$ requires the total area of the parts of the three circles on one side to be equal to the total area on the other side. Calculate the absolute value of the slope of this line.

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