
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 plane rectangular coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=\cos\theta}\\{y=1+\sin\theta}\end{array}}\right.$ ($\theta$ is the parameter), with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar equation of curve $C_{2}$ is $\rho ^{2}=2\rho \cos \theta +a$, where $a \gt -1$.  
$(1)$ When $a=0$, curve $C_{1}$ intersects curve $C_{2}$ at points $M$ and $N$. Find the length of the line segment $MN$.  
$(2)$ The line $l$ passing through the point $P(3,-1)$ has the parametric equations $\left\{{\begin{array}{l}{x=3+\frac{{\sqrt{2}}}{2}t,}\\{y=-1+\frac{{\sqrt{2}}}{2}t}\end{array}}\right.$ ($t$ is the parameter) and intersects curve $C_{2}$ at points $A$ and $B$. If $|PA|\cdot |PB|=1$, find the real number $a$.

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