
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 $C:\frac{{y}^{2}}{{a}^{2}}+\frac{{x}^{2}}{{b}^{2}}=1,\left( a > b > 0 \right)$, with an eccentricity of $\frac{\sqrt{2}}{2}$, foci $F_{1}(0,-c)$, $F_{2}(0,c)$, and a line passing through $F_{1}$ intersecting the ellipse at points $M$ and $N$. The perimeter of $\triangle F_{2}MN$ is $4$.

(I) Find the equation of the ellipse.

(II) A line $l:y=kx+m,\left( m\neq 0 \right)$ intersects the $y$-axis at point $P$ and intersects the ellipse $C$ at distinct points $A$ and $B$, such that $\overrightarrow{OA}+3\overrightarrow{OB}=4\overrightarrow{OP}$. Find the range of $m$.

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