
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
Consider the following sequence: 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4, 5/1, ..., where all fractions m/n with m+n = k are listed in order of decreasing m, and then all m/n with m+n = k+1, etc. Each rational appears many times. Find the first five positions of 1/2. What is the position for the nth occurrence of 1/2? Find an expression for the first occurrence of p/q where p < q and p and q are coprime.

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