
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 $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ be a point on one of the semicircular arcs determined by $\overline{MN}$ such that $MA= \frac{3}{4}$. Let $B$ be another point on the same arc such that $MB=\frac{1}{2}$. Point $C$ lies on the other semicircular arc at an equal distance from $M$ and $N$. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. The largest possible value of $d$ can be written in the form $r-s\sqrt{t}$, where $r, s$ and $t$ are positive integers, and $t$ is not divisible by the square of any prime. Find $r+s+t$.

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