
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
Consider the following system of equations for certain ordered pairs $(c,d)$ of real numbers:
\[\begin{aligned} cx+dy&=2 \\ x^2 + y^2 &= 65 \end{aligned}\]
Determine how many such ordered pairs $(c,d)$ exist such that the system has at least one solution, and each solution is an ordered pair $(x,y)$ of integers.

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