
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 $xOy$, the parametric equation of circle $C$ is $\begin{cases}x=a+\cos \theta \\ y=\sin \theta\end{cases}$ (where $\theta$ is the parameter). With the origin as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of line $l$ is $\rho\sin (\theta- \frac{\pi}{4})= \frac{ \sqrt{2}}{2}$. If line $l$ is tangent to circle $C$, then the real number $a=\boxed{-1\pm \sqrt{2}}$.

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