
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
$\triangle XYZ$ is inscribed inside $\triangle PQR$ such that $X, Y, Z$ lie on $QR, PR, PQ$, respectively. The circumcircles of $\triangle PYZ, \triangle QXZ, \triangle RXY$ have centers $O_1, O_2, O_3$, respectively. Given that $PQ = 26, QR = 28, PR = 27$, and $\stackrel{\frown}{QY} = \stackrel{\frown}{XZ},\ \stackrel{\frown}{QX} = \stackrel{\frown}{YZ},\ \stackrel{\frown}{PX} = \stackrel{\frown}{RY}$, the length of $QX$ can be written in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.

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