
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 \(ABCDA_1B_1C_1D_1\) has edge length 1. Point \(M\) is taken on the side diagonal \(A_1D\), and point \(N\) is taken on \(CD_1\), such that the line segment \(MN\) is parallel to the diagonal plane \(A_1ACC_1\). Find the minimum length of \(MN\).

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