
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
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is  $\tfrac{m+\sqrt{n}}{k}$  where  $m$ ,  $n$ , and  $k$  are integers, and  $k$  is not divisible by the square of any prime. Find  $m + n + k$ .

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