
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
Given $\forall x$, $y\in\mathbb{R}$, the inequality system
\[
\begin{cases}
& x^{2}+y^{2}+1 > 2bx+2y+b-b^{2}, \\
& 2\cos [2(x-y)]+8b\cos (x-y)+8b(b+1)+5 > 0 \\
\end{cases}
\]
always holds, determine the range of the real number $b$.

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