
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 real numbers \( a \) such that for any \( \theta \in \left[0, \frac{\pi}{2}\right] \), the following inequality holds:
$$(x + 3 + 2 \sin \theta \cos \theta)^2 + (x + a \sin \theta + a \cos \theta)^2 \geqslant \frac{1}{8}.$$

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