
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
A coordinate system and parametric equations (4-4)

In the rectangular coordinate system $xOy$, the parametric equations of curves $C_1$ and $C_2$ are given by $\begin{cases} x=4t^2 \\ y=4t \end{cases}$ ($t$ is a parameter) and $\begin{cases} x=\cos \varphi, \\ y=1+\sin \varphi \end{cases}$ ($\varphi$ is a parameter), respectively. Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.

(1) Find the Cartesian equation of curve $C_1$ and the polar equation of curve $C_2$.

(2) The ray $OM: \theta = \alpha$ ($\alpha \in [\frac{\pi}{6}, \frac{\pi}{4}]$) intersects curve $C_1$ at points $O$ and $P$, and intersects curve $C_2$ at points $O$ and $Q$. Find the maximum value of $|OP| \cdot |OQ|$.

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