
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
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=6$, $BC=8$, and $AC=5$. The median from vertex $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at another point $G$. Let $\gamma$ be the circle with diameter $\overline{DG}$. Circles $\omega$ and $\gamma$ meet at $G$ and a second point $H$. Find the squared length of segment $AH$.

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