
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 a tetrahedron \(ABCD\), the dihedral angle between the face \(ABC\) and the face \(BCD\) is \(60^\circ\). The projection of vertex \(A\) onto the plane \(BCD\), denoted as \(H\), is the orthocenter of \(\triangle BCD\). Let \(G\) be the centroid of \(\triangle ABC\). Given that \(AH = 4\) and \(AB = AC\), find the length \(GH\).

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