
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:
\[\begin{aligned} 
ax+by&=3 \\ 
x^2 + y^2 &= 29 
\end{aligned}\]
has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

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