
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$  and  $ B$  be points on one of the semicircular arcs determined by  $ \overline{MN}$  such that  $ A$  is the midpoint of the semicircle and  $ MB\equal{}\frac35$ . 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 the chords  $ \overline{AC}$  and  $ \overline{BC}$ . The largest possible value of  $ d$  can be written in the form  $ r\minus{}s\sqrt{t}$ , where  $ r$ ,  $ s$ , and  $ t$  are positive integers and  $ t$  is not divisible by the square of any prime. Find  $ r\plus{}s\plus{}t$ .

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