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

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