
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
Given the set $S$ as the points $(x, y)$ where $x \in \{0, 1, 2, 3, 4, 5\}$ and $y \in \{0, 1, 2, 3, 4\}$, excluding the point $(0,0)$. Consider $T$ to be the set of all right triangles whose vertices lie in $S$. For each right triangle $t = \triangle ABC$ with vertices $A, B, C$ (ordered counterclockwise) and a right angle at $C$, define $f(t) = \tan(\angle CAB)$. Calculate $\prod_{t \in T} f(t)$.

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