
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 $S = \{(x, y): x \in \{0, 1, 2, 3, 4\}, y \in \{1, 2, 3, 4, 5\}\}$. 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 a right angle at $A$, let $f(t) = \tan(\angle CBA)$. Suppose $S'$ is a similar set where $y$ values are from $\{0, 1, 2, 3, 4\}$. Construct an identical set $T'$ for $S'$ and consider the combined set $T''= T \cup T'$ with corresponding function $g(t)$ for triangles in $T'$. What is \[\prod_{t\in T''} (f(t)\cdot g(t))?\]
A) $1$
B) $\frac{625}{72}$
C) $\frac{150}{24}$
D) $5$
E) $\frac{625}{12}$

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