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

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