
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
Each edge of the triangular pyramid \( ABCD \) is equal to 1. Point \( E \) is on edge \( AB \), point \( F \) is on edge \( BC \), and point \( G \) is on edge \( CD \) such that \( AE = \frac{2}{3} \), \( BF = \frac{3}{4} \), and \( CG = \frac{1}{2} \). The plane \( EFG \) intersects line \( AD \) at point \( H \). Find the perimeter of triangle \( HEG \).

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