
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 that the edge length of the cube \( ABCD-A_1B_1C_1D_1 \) is 1, point \( M \) is chosen on the diagonal \( A_1D \) and point \( N \) is chosen on \( CD_1 \), such that segment \( MN \) is parallel to the diagonal face \( A_1ACC_1 \), calculate the minimum value of \( MN \).

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