
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 P be any point on the ellipse $\frac{x^2}{a} + \frac{y^2}{b} = 1$ (where $a > b > 0$), and let $F_1$ and $F_2$ be the foci of the ellipse, with $|PF_1| + |PF_2| = 4$ and the eccentricity being $\frac{\sqrt{3}}{2}$.  
(1) Find the standard equation of the ellipse.  
(2) Suppose the trajectory of point E is curve $C_1$, and the line $l: y = x + m$ intersects $C_1$ at points M and N. The perpendicular bisector of segment MN passes through point P(1, 0). Find the value of the real number $m$.

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