
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
 $A$ ,  $B$ ,  $C$ , and  $D$  are points on a circle, and segments  $\overline{AC}$  and  $\overline{BD}$  intersect at  $P$ , such that  $AP=8$ ,  $PC=1$ , and  $BD=6$ .  Find  $BP$ , given that  $BP<DP$ .

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