
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 $a, b \in \mathbb{R}$, $c \in [0, 2\pi)$, if for any real number $x$ the following equation holds true: $$2\sin(3x - \frac{\pi}{3}) = a\sin(bx + c)$$
Given the function $y=\sin^2x$ is defined on the interval $[0, 3\pi]$ and its graph intersects with the graph of $y=\cos x$. Denote the $x$-coordinate of their intersection points as $d$. The number of sets of ordered real numbers $(a, b, c, d)$ that satisfy these conditions is __________.

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