
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 $E: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ that passes through the point $(1, \frac {\sqrt {2}}{2})$ and has its foci at $(-1,0)$ and $(1,0)$:  
$(1)$ Find the equation of $E$;  
$(2)$ If $A$, $B$, and $P$ (where point $P$ does not coincide with the vertices of the ellipse) are three distinct points on $E$, and $O$ is the origin such that $\vec{OP}= \vec{OA}+ \vec{OB}$, find the minimum value of the area of the triangle formed by line $AB$ and the coordinate axes.

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