
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

\[
\left\{
\begin{array}{l}
x^{2}+y^{2}+2 a(a+y-x)=49 \\
y=15 \cos (x-b)-8 \sin (x-b)
\end{array}
\right.
\]

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

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