
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
Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle and $m\angle ABC = 45^\circ$. The length of hypotenuse $AB$ is $12$. Point $P$ is randomly chosen inside triangle $ABC$, and the line segment $\overline{BP}$ is extended to meet $\overline{AC}$ at $D$. What is the probability that the length of $BD > 6$?
A) $\frac{1}{2}$
B) $\frac{\sqrt{2} - 1}{\sqrt{2}}$
C) $\frac{2-\sqrt{2}}{2}$
D) $\frac{3-\sqrt{3}}{3}$
E) $\frac{5-\sqrt{5}}{5}$

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