
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 $F_1$, $F_2$ are the two foci of the ellipse $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$, $O$ is the origin, and point $P(-1, \frac { \sqrt {2}}{2})$ is on the ellipse. The line segment $PF_2$ intersects the y-axis at point $M$ satisfying $\overrightarrow {PM}+ \overrightarrow {F_{2}M}= \overrightarrow {0}$. $\bigcirc O$ is the circle with diameter $F_1F_2$, and a line $L: y=kx+m$ is tangent to $\bigcirc O$ and intersects the ellipse at two distinct points $A$ and $B$.
(1) Find the standard equation of the ellipse.
(2) When $\overrightarrow {OA}\cdot \overrightarrow {OB}=\lambda$ and $\frac {2}{3}≤\lambda≤ \frac {3}{4}$, find the range of the area $S$ of $\triangle AOB$.

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