
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 defined by $\begin{cases} x=a+a\cos \phi \\ y=a\sin \phi \end{cases}$ (with $\phi$ as the parameter, real number $a > 0$), and the curve $C_{2}$ is defined by $\begin{cases} x=b\cos \phi \\ y=b+b\sin \phi \end{cases}$ (with $\phi$ as the parameter, real number $b > 0$). In the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the ray $l$: $\theta=\alpha (\rho \geqslant 0, 0 \leqslant \alpha \leqslant \frac{\pi}{2})$ intersects $C_{1}$ at points $O$ and $A$, and intersects $C_{2}$ at points $O$ and $B$. When $\alpha=0$, $|OA|=1$; when $\alpha= \frac{\pi}{2}$, $|OB|=2$.  
(Ⅰ) Find the values of $a$ and $b$;  
(Ⅱ) Find the maximum value of $2|OA|^{2}+|OA|\cdot|OB|$.

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