
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
Given a tetrahedron \(ABCD\), a sphere with a radius of 1 is drawn through vertex \(D\). This sphere is tangent to the circumsphere of the tetrahedron at point \(D\) and also tangent to the face \(ABC\). Given that \(AD=2\sqrt{3}\), \(\angle BAD = \angle CAD = 45^\circ\), and \(\angle BAC = 60^\circ\), find the radius \(r\) of the circumsphere of the tetrahedron.

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