
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
a. Find all integer values of \(a\) such that equation \(x^2 + ax + 1 = 0\) does not have real solutions in \(x\).

b. Find all pairs of integers \((a, b)\) such that both equations
\[ x^2 + ax + b = 0 \quad \text{and} \quad x^2 + bx + a = 0 \]
have no real solutions in \(x\).

c. How many ordered pairs \((a, b)\) of positive integers satisfying \(a \leq 8\) and \(b \leq 8\) are there, such that each of the equations
\[ x^2 + ax + b = 0 \quad \text{and} \quad x^2 + bx + a = 0 \]
has two unique real solutions in \(x\)?

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