
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
$4-4$: Polar Coordinates and Parametric Equations

The polar coordinate system and the rectangular coordinate system $(xOy)$ have the same unit of length, with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis $(.)$. The polar coordinate equation of the curve $C_{1}$ is given by $ρ=2 \sqrt {2}\sin (θ+ \dfrac {π}{4})$, and the polar coordinate equation of the curve $C_{2}$ is given by $ρ\sin θ=a(a > 0)$. The rays $θ=φ,θ=φ+ \dfrac {π}{4},θ=φ- \dfrac {π}{4}$, $θ= \dfrac {π}{2}+φ$ intersect the curve $C_{1}$ at four points $A$, $B$, $C$, $D$ respectively, distinct from the pole $O$.

$(\text{I})$ If the curve $C_{1}$ is symmetric about the curve $C_{2}$, find the value of $a$, and convert the curves $C_{1}$ and $C_{2}$ into rectangular coordinate equations.

$(\text{II})$ Find the value of $|OA|⋅|OC|+|OB|⋅|OD|$.

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