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

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