
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 $xOy$, there is a curve $C_1$ defined parametrically as $C_1: \begin{cases} x=1+\cos\theta \\ y=\sin\theta\end{cases}$ (where $\theta$ is a parameter). The point $P$ is the intersection of the curve $C_1$ with the positive half of the x-axis. In the polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis, there is a curve $C_2$ defined by the equation $\rho\cos\theta+\rho\sin\theta+3=0$.
1. Find the polar equation of curve $C_1$ and the polar equation of the tangent line to curve $C_1$ at point $P$.
2. Find a point $Q(a, b)$ on curve $C_1$ such that its distance to curve $C_2$ is the longest.

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