
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, a polar coordinate system is established. The polar equations of circle $C_1$ and line $C_2$ are respectively $\rho=4\sin \theta$ and $\rho \cos \left( \theta -\frac{\pi}{4} \right)=2\sqrt{2}$.

$(1)$ Find the polar coordinates of the intersection points of $C_1$ and $C_2$;

$(2)$ 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 $\left\{ \begin{array}{l} x=t^3+a, \\ y=\frac{b}{2}t^3+1 \end{array} \right.$ (where $t \in \mathbb{R}$ is the parameter), find the values of $a$ and $b$.

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