
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
Given points $A(2,0)$, $B(0,6)$, and $O$ as the origin. If $D$ is the symmetric point of $O$ with respect to the line $AB$, and $BD$ is extended to point $P$ such that $|PD|=2|BD|$. If the line $l:ax+10y+84-108\sqrt{3}=0$ passes through point $P$, find the slope of line $l$.

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