
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
In triangle $\triangle ABC$, with internal angles $A$, $B$, $C$, and $0 \lt B\leqslant \frac{π}{3}$. If for any real number $x$, the inequality $\left(x+2+\sin 2B\right)^{2}+[\sqrt{2}t\cdot \sin (B+\frac{π}{4})]^{2}\geqslant 1$ always holds, find the range of the real number $t$.

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