
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 polar coordinate system Ox, curve $C_1$: $\rho\cos\left(\theta+ \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, curve $C_2$: $\rho=1$ ($0\leq\theta\leq\pi$), and curve $C_3$: $\frac{1}{\rho^2} = \frac{\cos^2\theta}{3}+\sin^2\theta$. Let curve $C_1$ and $C_2$ intersect at point M.  
(I) Find the polar coordinates of point M.  
(II) If a moving line $l$ passes through point M and intersects curve $C_3$ at two distinct points A and B, find the minimum value of $\frac{|MA|\cdot|MB|}{|AB|}$.

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