
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, consider the system of equations
\[\begin{aligned} ax+by &= 1 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs \( (a,b) \) exist, where each solution \( (x,y) \) is an ordered pair of integers.

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