
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
$\triangle GHI$ is inscribed inside $\triangle JKL$ such that $G, H, I$ lie on $KL, JL, JK$, respectively. The circumcircles of $\triangle GIC, \triangle HLG, \triangle IJH$ have centers $P_1, P_2, P_3$, respectively. Additionally, $JK = 20, KL = 27, JL= 25$, and $\stackrel{\frown}{HG} = \stackrel{\frown}{IC},\ \stackrel{\frown}{JH} = \stackrel{\frown}{GC},\ \stackrel{\frown}{IJ} = \stackrel{\frown}{GL}$. The length of $GL$ can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.

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