
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 \( S = \{(x,y) : x \in \{0,1,2,3,4,5\}, y \in \{0,1,2,3\},\text{ and } (x,y)\ne (5,3)\} \). 
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 \( f(t) = \tan(\angle{CBA}) \). Find the product of $f(t)$ for every $t$ in $T$.

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