
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
Let the vectors \(\boldsymbol{\alpha} = (x + 3, x)\) and \(\boldsymbol{\beta} = (2 \sin \theta \cos \theta, a \sin \theta + a \cos \theta)\) satisfy \(|\boldsymbol{\alpha} + \boldsymbol{\beta}| \geq \sqrt{2}\) for any \(x \in \mathbb{R}\) and \(\theta \in \left[0, \frac{\pi}{2}\right]\). Find the range of the real number \(a\).

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