
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
How many distinct triangles  $ABC$  are tjere, up to simplilarity, such that the magnitudes of the angles  $A, B$  and  $C$  in degrees are positive integers and satisfy  $$ \cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1 $$  for some positive integer  $k$ , where  $kC$  does not exceet  $360^{\circ}$ ?

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