
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ ($a > b > 0$), and point $A$($1, \frac{3}{2}$) on the ellipse $C$ has a distance sum of $4$ to $F\_1$ and $F\_2$. Determine the equation of the ellipse $C$ and the coordinates of its foci.

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