
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
In triangle $ABC,$ $G$ lies on $\overline{AB}$ such that $AG:GB = 3:2,$ and $H$ lies on $\overline{BC}$ such that $BH:HC = 1:3$. Let $D$ be the intersection of $\overline{AG}$ and $\overline{CH}.$ Similarly, if $K$ lies on $\overline{AC}$ where $AK:KC = 1:2$, and $L$ lies on $\overline{BC}$ such that $BL:LC = 2:1$, let $J$ be the intersection of $\overline{DL}$ and $\overline{BK}.$ Determine the vector $\overrightarrow{J}$ in barycentric coordinates $(x, y, z)$ relative to $\overrightarrow{A}$, $\overrightarrow{B}$, $\overrightarrow{C}$ where $x + y + z = 1$.

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