
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 Cartesian coordinate system $xOy$, $\Gamma_{1}$ is a unit circle centered at $(2,1)$, and $\Gamma_{2}$ is a unit circle centered at $(10,11)$. A line $l$ passes through the origin $O$ and intersects each of $\Gamma_{1}$ and $\Gamma_{2}$ at two points, dividing $\Gamma_{1}$ and $\Gamma_{2}$ into four circular arcs. Among these four circular arcs, two are of equal length. The sum of the slopes of all lines $l$ that satisfy these conditions is ______.

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