
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
(The full score of this question is 14 points) Given the function $f(x) = 4x^3 - 3x^2\cos\theta + \cos\theta$, where $x \in \mathbb{R}$, $\theta$ is a parameter, and $0 \leq \theta \leq 2\pi$.
1. When $\cos\theta = 0$, determine whether the function $f(x)$ has an extremum;
2. To make the local minimum of the function $f(x)$ greater than zero, find the range of the parameter $\theta$;
3. If for any parameter $\theta$ within the range found in (2), the function $f(x)$ is increasing in the interval $(2a - 1, a)$, find the range of the real number $a$.

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