
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 $ABC$, side lengths are $AB = 3$, $BC = 4$, and $CA = 5$. Points $G$ and $H$ are on ray $AB$ such that $AB < AG < AH$. Another point $I \neq C$ is the point of intersection of the circumcircles of $\triangle ACG$ and $\triangle BHC$, satisfying $GI = 3$ and $HI = 8$. Determine the length of $BH$ expressed as $\tfrac{p+q\sqrt{r}}{s}$, where $p$, $q$, $r$, and $s$ are positive integers with $p$ and $s$ relatively prime, and $r$ not divisible by the square of any prime. Find $p+q+r+s$.

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