
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
Let ellipse C: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\; (a>b>0)$ have the left and right foci F<sub>1</sub>, F<sub>2</sub>, and the moving straight line l passing through F<sub>1</sub> intersects ellipse C at points A and B. It is known that the perimeter of triangle F<sub>2</sub>AB is 8 and the point P $(1,\frac{3}{2})$ lies on ellipse C.  
(1) Find the standard equation of ellipse C;  
(2) Does there exist a fixed point M on the x-axis such that the angle ∠F<sub>1</sub>MA=∠F<sub>1</sub>MB always holds? If yes, find the coordinates of point M; if not, explain the reason.

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