
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
The product of four distinct numbers, which are the roots of the equations

$$
x^{2} + 6bx + c = 0 \quad \text{and} \quad x^{2} + 6cx + b = 0
$$

is equal to 1. At least one of the numbers \( b \) and \( c \) is an integer. Find \( b + c \) given that \( b \) and \( c \) are positive.

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