
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 equation $\sqrt{3}\sin 2x + \cos 2x = k + 1$ to have a real root within the interval $\left[0, \frac{\pi}{2}\right]$, determine the range of $k$.

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