
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 rectangular parallelepiped \( ABCDA_1B_1C_1D_1 \) (\( AA_1 \parallel BB_1 \parallel CC_1 \parallel DD_1 \)), it is known that \( AB = BC = 2a \), and \( AA_1 = a \). A plane of section passes through the points \( B_1 \) and \( D \) parallel to the line \( AC \). Find the radius of the sphere that touches this section and three faces of the parallelepiped with the common vertex \( B \).

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