
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 \(ABCD A_{1} B_{1} C_{1} D_{1}\), a plane passing through \(A\) and tangent to the sphere inscribed in the cube intersects the edges \( A_{1} B_{1} \) and \( A_{1} D_{1} \) at points \( K \) and \( N \) respectively. Determine the measure of the dihedral angle between the planes \( AC_{1}K \) and \( AC_{1}N \).

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