
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
Determine the number of ordered pairs $(b,c)$ of positive integers such that neither $x^2 + 2bx + c = 0$ nor $x^2 + 2cx + b = 0$ have two distinct real solutions, and both $b$ and $c$ are less than or equal to $10$.
A) 3
B) 4
C) 5
D) 6
E) 7

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