
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
In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is given by the parametric equations 
\[
\begin{cases}
x = t \cos \alpha, \\
y = 1 + t \sin \alpha
\end{cases}
(t \text{ is the parameter, } 0 \leq \alpha < \pi).
\]
A polar coordinate system is established with the origin as the pole and the positive $x$-axis as the polar axis. The polar equation of the curve $C_{2}$ is 
\[
\rho + \frac{7}{\rho} = 4\cos \theta + 4\sin \theta.
\]
(Ⅰ) When $\alpha = \frac{\pi}{2}$, write down directly the ordinary equation and polar equation of $C_{1}$, and the ordinary equation of $C_{2}$.
(Ⅱ) Given the point $P(1, \frac{\pi}{2})$, and the curves $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Find the value of $|PA| \cdot |PB|$.

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