
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 function $f(x) = 4\cos x \sin\left(x + \frac{\pi}{6}\right) - 1$,  
(I) find the increasing intervals for $f(x)$;  
(II) if $\sin^2x + af\left(x + \frac{\pi}{6}\right) + 1 > 6\cos^4x$ holds true for any $x \in \left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$, determine the range of the real number $a$.

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