
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 $a \in \mathbb{R}$, let $f(x) = \cos x (a \sin x - \cos x) + \cos^2 (\frac{\pi}{2} - x)$ satisfy $f(-\frac{\pi}{3}) = f(0)$.
(I) Find the interval where the function $f(x)$ is strictly increasing;
(II) Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and satisfy $\frac{a^2 + c^2 - b^2}{a^2 + b^2 - c^2} = \frac{c}{2a - c}$. Find the range of $f(x)$ on $(0, B]$.

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