
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
Through the diagonal \( B_1D_1 \) of the face \( A_1B_1C_1D_1 \) and the midpoint of the edge \( DC \) of a regular quadrilateral prism \( ABCD \, A_1B_1C_1D_1 \), a plane is drawn. Find the area of the cross-section of the prism by this plane, given that \( AB = a \) and \( CC_1 = 2a \).

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