
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 that $1 < m < 4$, $F\_1$ and $F\_2$ are the left and right foci of the curve $C: \frac{x^2}{4} + \frac{y^2}{4-m} = 1$, and point $P$ is the intersection point of curve $C$ and curve $E: x^2 - \frac{y^2}{m-1} = 1$ in the first quadrant. Line $l$ is the tangent line to curve $C$ at point $P$. If the incenter of triangle $F\_1 P F\_2$ is point $M$, and line $F\_1 M$ intersects line $l$ at point $N$, find the sum of the x-coordinates of points $M$ and $N$.

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