
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 $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ (where $a>b>0$) with eccentricity $e=\frac {\sqrt {2}}{2}$, point $P(1,\frac {\sqrt {2}}{2})$ lies on this ellipse.

(1) Find the standard equation of the ellipse.
(2) If a line $l$ is tangent to the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse at two distinct points $A$ and $B$, find the maximum value of the area $S$ of triangle $\triangle AOB$.

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