
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 functions $f(x) = \sin{2x} - 1$ and $g(x) = 2a(\sin{x} + \cos{x}) - 4ax$. If there exists some $x \in \left[0, \frac{\pi}{2}\right]$ such that $f(x) \geqslant g''(x)$, find the range of the real number $a$.

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