
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
Given two strictly increasing sequences of positive integers $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ that satisfy $a_{10} = b_{10} < 2017$, with the conditions $a_{n+2} = a_{n+1} + a_{n}$ and $b_{n+1} = 2b_{n}$ for any positive integer $n$, find all possible values of $a_{1} + b_{1}$.

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