
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 tetrahedron \(ABCD\), \(AD = 2\sqrt{3}\), \(\angle BAC = 60^\circ\), \(\angle BAD =\) \(\angle CAD = 45^\circ\); if a sphere tangent to plane \(ABC\) at point \(D\) and inscribed within the circumsphere of the tetrahedron has a radius of 1, find the radius of the circumsphere of tetrahedron \(ABCD\).

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