
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
Let $\overline{PQ}$ be a diameter of a circle with diameter 2. Let $X$ and $Y$ be points on one of the semicircular arcs determined by $\overline{PQ}$ such that $PX=\frac{4}{5}$ and $PY=\frac{3}{4}$. Point $Z$ lies on the other semicircular arc. Let $e$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{PQ}$ with chords $\overline{XZ}$ and $\overline{YZ}$. The largest possible value of $e$ can be written in the form $u-v\sqrt{w}$, where $u, v$ and $w$ are positive integers and $w$ is not divisible by the square of any prime. Find $u+v+w$.

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