
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
Let  $P$  be the portion of the graph of  $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$  located in the first quadrant (not including the  $x$  and  $y$  axes). Let the shortest possible distance between the origin and a point on  $P$  be  $d$ . Find  $\lfloor 1000d \rfloor$ .

*Proposed by **Th3Numb3rThr33***

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