
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 expansion of \((x + 2)^n\) where three consecutive terms have coefficients in the ratio \(1:3:5\). These can be expressed as \[{n\choose k} : {n\choose k+1} : {n \choose k+2}\]. Determine the sum of all possible values of \(n+k\).

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