
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 curve $C_1$ with the parametric equation \[\begin{cases}x=1+\cos \theta,\\ y=1+\sin \theta\end{cases}\] ($\theta$ is the parameter), establish a polar coordinate system with the origin as the pole and the non-negative half-axis of $x$ as the polar axis. The curve $C_2$ is described by its polar equation $\rho=1$. Convert the parametric equation of curve $C_1$ into a standard equation and the polar equation of curve $C_2$ into a Cartesian coordinate equation.

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