
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 the inequality \(\sqrt{2}(2a + 3) \cos \left(\theta - \frac{\pi}{4} \right) + \frac{6}{\sin \theta + \cos \theta} - 2 \sin 2\theta < 3a + 6\) holds for all \(\theta \in \left[0, \frac{\pi}{2} \right]\), find the range of \(a\).

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