
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 an ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ ($a>b>0$) with left and right foci $F_1$ and $F_2$ respectively, the focal distance is $2\sqrt{3}$. Point $P$ is on the ellipse, $\angle F_1PF_2 = 90^\circ$, and the area of $\triangle F_1PF_2$ is 1.
(I) Find the standard equation of the ellipse;
(II) Let point $B$ be the upper vertex of the ellipse. A line $l$ passes through an inner point $M(0, m)$ of the ellipse and intersects the ellipse at points $C$ and $D$. If the area ratio of $\triangle BMC$ to $\triangle BMD$ is 2:1, find the range of real number $m$.

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