
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 \(\Gamma: x^{2}=4 y\) with focus \(F\), two different lines \(l_{1}\) and \(l_{2}\) pass through \(F\) 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 \(B\), and line \(l_{2}\) intersects the parabola \(\Gamma\) at points \(C\) and \(D\). The lines \(AB\) and \(CD\) are diameters of circles \(\odot M\) and \(\odot N\) respectively, and the chord through the center of these circles is denoted as line \(l\). Find the minimum distance from point \(M\) (the center of \(\odot M\)) to line \(l\).

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