
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
Find all pairs of positive integers  $ a,b$  such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a  a^2+a+1 &\equiv  0 \pmod b . \end{align*}

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