
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 $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equations of circle $C_{1}$ and line $C_{2}$ are respectively $\rho=4\sin \theta$ and $\rho\cos (\theta- \frac {\pi}{4})=2 \sqrt {2}$.  
(Ⅰ) Find the polar coordinates of the intersection points of $C_{1}$ and $C_{2}$.  
(Ⅱ) Let $P$ be the center of circle $C_{1}$, and $Q$ be the midpoint of the line connecting the intersection points of $C_{1}$ and $C_{2}$. Given that the parametric equation of line $PQ$ is $\begin{cases} x=t^{3}+a \\ y= \frac {b}{2}t^{3}+1 \end{cases}(t\in\mathbb{R})$ with $t$ as the parameter, find the values of $a$ and $b$.

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