
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 expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:3:5$ that can be written in the form \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]. Find the sum of all possible values of $n+k$.

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