
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
$\triangle KLM$ is inscribed inside $\triangle XYZ$ such that $K, L, M$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle KYZ, \triangle LXK, \triangle MLY$ have centers $P_1,P_2,P_3$, respectively. Also, $XY = 24, YZ = 29, XZ = 27$, and $\stackrel{\frown}{MY} = \stackrel{\frown}{LZ},\ \stackrel{\frown}{MX} = \stackrel{\frown}{KY},\ \stackrel{\frown}{LX} = \stackrel{\frown}{KZ}$. The length of $KY$ can be written in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers. Find $a+b$.

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