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

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