
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 the right focus $F\_1(1,0)$ of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2}=1 (a>b>0)$ and the eccentricity $\frac{1}{2}$:
(I) Find the equation of the ellipse $C$ and the coordinates of the left vertex $P$;
(II) Suppose the line passing through $F\_1$ intersects the ellipse $C$ at points $A$ and $B$. If the area of $\triangle PAB$ is $\frac{36}{13}$, find the equation of the line $AB$.

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