
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
Triangle  $ ABC$  is inscript in a circumference  $ \Gamma$ . A chord  $ MN\equal{}1$  of  $ \Gamma$  intersects the sides  $ AB$  and  $ AC$  at  $ X$  and  $ Y$  respectively, with  $ M$ ,  $ X$ ,  $ Y$ ,  $ N$  in that order in  $ MN$ . Let  $ UV$  be the diameter of  $ \Gamma$  perpendicular to  $ MN$  with  $ U$  and  $ A$  in the same semiplane respect to  $ MN$ . Lines  $ AV$ ,  $ BU$  and  $ CU$  cut  $ MN$  in the ratios  $ \frac{3}{2}$ ,  $ \frac{4}{5}$  and  $ \frac{7}{6}$  respectively (starting counting from  $ M$ ). Find  $ XY$ 

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