
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 a cube $A B C D A_1 B_1 C_1 D_1$ with bases $A B C D$ and $A_1 B_1 C_1 D_1$. Point $M$ is the midpoint of edge $A B$, and $K$ is the midpoint of edge $C D$. Find the radius of the sphere passing through points $M, K, A_1, C_1$, if the edge length of the cube is $\sqrt{41}$.

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