
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, with the origin as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that curve $C_{1}$: $\rho=\cos \theta-\sin \theta$, and curve $C_{2}$ is given by $\begin{cases}x= \frac{1}{2}- \frac{ \sqrt{2}}{2}t \\ y= \frac{ \sqrt{2}}{2}t\end{cases}$ (where $t$ is a parameter).

$(1)$ Find the rectangular coordinate equation of curve $C_{1}$;  
$(2)$ If curve $C_{1}$ intersects with curve $C_{2}$ at points $P$ and $Q$, find the value of $|PQ|$.

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