
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 semicircular arc determined by $\overline{PQ}$ such that $X$ is one quarter of the way along the semicircle from $P$ and $PY=\frac{2}{5}$. Point $Z$ lies on the other semicircular arc, whereas the points of intersection of $\overline{PQ}$ with chords $\overline{XZ}$ and $\overline{YZ}$ are $R$ and $S$ respectively. Calculate the greatest possible value of the length $e$ of the segment $\overline{RS}$.

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