
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 $b$ for which the inequality $x^{2}+6x+2(a+b+1)\sqrt{-x^{2}-6x-5}+8 < a^{2}+b^{2}+2a$ is not satisfied for at least one value of $x \in [-5 ; -1]$ for any value of the parameter $a \in [-1 ; 1]$.

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