
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 side length of the base \( ABC \) of a regular pyramid \( ABCD \) is \( 4 \sqrt{3} \), and \( \angle DAB = \operatorname{arctg} \sqrt{\frac{37}{3}} \). Points \( A_1 \), \( B_1 \), and \( C_1 \) are the midpoints of the edges \( AD \), \( BD \), and \( CD \) respectively. Find:
1) The angle between the lines \( BA_1 \) and \( AC_1 \).
2) The distance between the lines \( BA_1 \) and \( AC_1 \).
3) The radius of the sphere that touches the plane \( ABC \) and the segments \( AC_1 \), \( BA_1 \), and \( CB_1 \).

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