
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 1. Let $X$ and $Y$ be points on one of the semicircular arcs determined by $\overline{PQ}$ such that $X$ is the midpoint of the semicircle and $PY=\frac{2}{5}$. 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 $\overline{YZ}$. Find the largest possible value of $e$, which can be expressed 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.

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