
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
Given that the discriminants of the quadratic equations $x^2+bx+c=0$ and $x^2+cx+b=0$ satisfy $b^2 - 9c \leq 0$ and $c^2 - 9b \leq 0$, and both equations have no two distinct real solutions, find the number of ordered pairs $(b,c)$ of positive integers.

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