
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 the range of the real number \( a \) such that the inequality \( \sin 2\theta - 2\sqrt{2} a \cos\left(\theta-\frac{\pi}{4}\right) - \frac{\sqrt{2}a}{\sin\left(\theta+\frac{\pi}{4}\right)} > -3 - a^2 \) holds for all \( \theta \in \left[0, \frac{\pi}{2}\right] \).

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