
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 a right triangle $ABC$, $\angle ACB = 90^\circ$, $\angle ABC = 45^\circ$, and the hypotenuse $AB = 12$. Let point $P$ be randomly chosen inside triangle $ABC$, and extend line segment $\overline{BP}$ to meet $\overline{AC}$ at point $D$. Determine the probability that $BD > 6\sqrt{2}$ and $P$ lies above the median line of $BC$.

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