
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 $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) passing through the point $(1, \frac {\sqrt {3}}{2})$, the left and right vertices of the ellipse are $A_1$ and $A_2$ respectively, the coordinates of point $P$ are $(4, 0)$, and the lengths $|PA_1|$, $|A_1A_2|$, $|PA_2|$ form an arithmetic sequence.

1. Find the standard equation of the ellipse.
2. Determine if there is a fixed point inside the ellipse such that any line passing through this point intersects the ellipse at points $M$ and $N$, and the condition $\overrightarrow {PM} \cdot \overrightarrow {PN} = 12$ always holds. If such a point exists, find it; if not, explain why.

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