
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 the parabola $\Gamma: x^{2}=4y$ with focus $F_{0}$, two different lines $l_{1}$ and $l_{2}$ pass through $F_{0}$ with slopes $k_{1}$ and $k_{2}$ respectively, such that $k_{1} + k_{2} = 2$. Line $l_{1}$ intersects the parabola $\Gamma$ at points $A$ and $B$, while line $l_{2}$ intersects the parabola $\Gamma$ at points $C$ and $D$. Denote the circles with diameters $AB$ and $CD$ as $\odot M$ and $\odot N$, respectively. The common chord of these two circles lies on a line $l$. Find the minimum distance from point $M$ to line $l$.

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