
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 2. Let $A$ be the midpoint of the semicircular arc determined by $\overline{MN}$ and let $MB = \frac{4}{7}$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with chords $\overline{AC}$ and $\overline{BC}$. Find the largest possible value of $d$, expressing your answer as $r - s\sqrt{t}$ where $r, s,$ and $t$ are integers.

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