
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
Given the ellipse $C: \frac{x^{2}}{4m}+ \frac{y^{2}}{m}=1(m>0)$ intersects with the x-axis at two points $A_1$, $A_2$, and with the y-axis at one point $B$, the area of $\triangle BA_1A_2$ is 2.
(I) Find the equation of ellipse C and its eccentricity;
(II) On the right side of the y-axis and parallel to it, a line $l$ intersects the ellipse at two distinct points $P_1$, $P_2$. Lines $A_1P_1$ and $A_2P_2$ intersect at point $P$. With the origin $O$ as the center and $A_1B$ as the radius, a circle intersects the x-axis at two points $M$, $N$ (with $M$ on the left side of $N$). Find the value of $|PM|-|PN|$.

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