
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
A cube \( ABCD A_1 B_1 C_1 D_1 \) is given. A plane passing through the vertex \( A \) 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 \( N \). 

Find the angle between the planes \( AC_1 K \) and \( AC_1 N \).

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