
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
1. In the Cartesian coordinate system $xOy$, the parametric equation of circle $C$ is $\begin{cases} x=1+\cos φ \ y=\sin φ \end{cases}$ ($φ$ is the parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.

   a. Find the polar equation of circle $C$.

   b. Given that the polar equation of line $l$ is $ρ(\sin θ+ \sqrt{3}\cos θ)=3 \sqrt{3}$, the ray $OM$: $θ= \dfrac{π}{3}$ intersects with circle $C$ at points $O$ and $P$, and with line $l$ at point $Q$. Find the length of line segment $PQ$.

2. Given the function $f(x)=x+1+|3-x|$, $x\geqslant -1$.

   a. Find the solution set of the inequality $f(x)\leqslant 6$.

   b. If the minimum value of $f(x)$ is $n$, and positive numbers $a$ and $b$ satisfy $2nab=a+2b$, find the minimum value of $2a+b$.

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