Diversity of Though

[State reasoning problem here for example: In the following question, a number series is given with one term missing. Choose the correct alternative that will follow the same pattern and fill in the blank spaces. 1, 2, 3, 5, x, 13]

To begin, please suggest 3 distinct approaches I could use to accurately solve the above problem:

Approach 1:
Approach 2:
Approach 3:
Now please provide 3 short demonstrations, each solving the original problem using one of the approaches you suggested above:

Demonstration 1 (Approach 1):

Demonstration 2 (Approach 2):

Demonstration 3 (Approach 3):

Great, let's put it all together. Please now take on the role of expert one (a persona you feel is mostly aligned to the issue) and solve the original problem using Approaches 1-3.

Now take on the persona of expert 2 (a persona you feel is the next most likely aligned to the issue) and solve the original problem again using Approaches 1-3.

Finally, take on the persona of expert 3 (a persona you feel is the next most likely aligned to the issue) and solve the original problem a third time using Approaches 1-3.

Please synthesize your responses from the 3 expert personas above and provide your final recommended solution.