
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
The medians $A E$, $B F$, and $C D$ of triangle $ABC$ are drawn, along with segment $F H$, which is parallel and equal to $A E$, and point $G$ is the midpoint of $B H$. The points $E$, $F$, and $G$ are collinear. The following statements are made about the given diagram:

1. $H E = H G$
2. $B H = D C$
3. $F G = \frac{3}{4} A B$.

Are all these statements true, and why?

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