
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
For the inequality $\sqrt{2}(2a + 3) \cos \left(\theta - \frac{\pi}{4}\right) \div \frac{6}{\sin \theta + \cos \theta} \geq 2 \sin 2\theta - 3a + 6$, for $\theta \in \left[ 0, \frac{\pi}{2} \right)$ to be true for all $\theta$ in this interval, find the possible values of $a$.

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