
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
The edge of cube \( ABCD A_1 B_1 C_1 D_1 \) is equal to \( a \). Find the radius of the sphere passing through the midpoints of the edges \( AA_1, BB_1 \) and through the vertices \( A \) and \( C_1 \).

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