
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
Find, with proof, all ways to write 1 as a sum of three fractions, each with numerator 1 and positive integer denominator. (The order of the fractions is irrelevant, so for instance \(\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\) is the same as \(\frac{1}{4}+\frac{1}{4}+\frac{1}{2}\).)

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