
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 $b$ and $c$ are positive integers, find the number of ordered pairs $(b,c)$ that satisfy the conditions: the equation $x^2+bx+c=0$ has exactly one real solution, and the equation $x^2+cx+b=0$ has no two distinct real solutions, with the additional constraint that $b + c \leq 10$.

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