
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 triangle \( \triangle ABC \), let \( D \) and \( E \) be the trisection points of \( BC \), with \( D \) between \( B \) and \( E \). Let \( F \) be the midpoint of \( AC \), and \( G \) be the midpoint of \( AB \). Let \( H \) be the intersection point of segments \( EG \) and \( DF \). Find the ratio \( EH : HG \).

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