
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
For certain ordered pairs \( (a,b) \) of real numbers, the system of equations
\[
\begin{aligned}
ax+by &= 2 \\
x^2 + y^2 &= 65
\end{aligned}
\]
has at least one solution, and each solution is an ordered pair \( (x,y) \) of integers. Determine how many such ordered pairs \( (a,b) \) exist.

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