
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
Find all values of the parameter $a$ for each of which there exists a number $b$ such that the system

$$
\left\{\begin{array}{l}
x=\frac{6}{a}-|y-a| \\
x^{2}+y^{2}+b^{2}+63=2(b y-8 x)
\end{array}\right.
$$

has at least one solution $(x, y)$.

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