
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
In the cube \( ABCD - A_1B_1C_1D_1 \) with a side length of 2, \( M \) and \( N \) are the midpoints of edges \( BB_1 \) and \( B_1C_1 \) respectively. If \( P \) is a point moving within the plane \( DMN \), and if the distance from point \( P \) to the plane \( BCC_1B_1 \) is equal to the length of \( PD \), what is the eccentricity of the trajectory of point \( P \)?

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