
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
In the Cartesian coordinate system $(xOy)$, a polar coordinate system is established with the coordinate origin $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis. The polar coordinate equation of the curve $C$ is given by $({{\rho }^{2}}=1+2\rho \cos \theta )$.

(I) Convert the polar coordinate equation of curve $C$ into the ordinary equation in the rectangular coordinate system;

(II) Line $l$ passes through the fixed point $P(1, \frac{1}{2})$, and intersects curve $C$ at points $A$ and $B$. Find the maximum value of $| \frac{1}{|PA|} - \frac{1}{|PB|} |$.

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