
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 right triangle $ABC$, with a right angle at $B$, point $D$ is the foot of the altitude from $B$ onto $AC$. Given $AD=4$ and $DC=5$, find the area of triangle $ABC$ and the length of segment $BD$.
**A)** $9\sqrt{5}$, $2\sqrt{5}$
  **B)** $18\sqrt{5}$, $4\sqrt{5}$
  **C)** $45$, $10$
  **D)** $4.5\sqrt{5}$, $2\sqrt{5}$
  **E)** $4.5\sqrt{5}$, $4\sqrt{5}$

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