
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 $S=\{(x,y): x\in \{0,1,2,3,4,5\}, y\in \{0,1,2,3,4,5\}, \text{ and } (x,y) \ne (0,5)\}$. Let $T$ be the set of all right triangles whose vertices are points in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order, and a right angle at $B$, define $f(t) = \sin(\angle ABC) \cdot \tan(\angle CAB)$. Find the product $\prod_{t \in T} f(t)$.

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