
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
Let \( S=\{(x,y) : x\in \{0,1,2,3\}, y\in \{0,1,2,3,4\},\text{ and } (x,y)\ne (0,4)\} \). 
Let \( T \) be the set of all right triangles whose vertices are in \( S \). For every right triangle \( t=\triangle{ABC} \) with vertices \( A \), \( B \), and \( C \) in counter-clockwise order and right angle at \( A \), let \( f(t)=\tan(\angle{CBA}) \). What is \(\prod_{t\in T} f(t)?\)
A) \(\frac{9}{2}\)
B) \(\frac{3}{4}\)
C) \(3\)
D) \(4\)
E) \(6\)

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