
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
Two lines AB and CD pass through a point P outside two parallel planes $\alpha$ and $\beta$. They intersect plane $\alpha$ at points A and C, respectively, and intersect plane $\beta$ at points B and D, respectively. Given that PA = 6, AC = 9, and PB = 8, the length of BD is \_\_\_\_\_\_\_.

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