
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 \( b \) such that the system

$$
\left\{\begin{array}{l}
x \cos a + y \sin a + 3 \leq 0 \\
x^2 + y^2 + 8x - 4y - b^2 + 6b + 11 = 0
\end{array}\right.
$$

has at least one solution for any value of the parameter \( a \).

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