
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. Given that ${(3x-2)^{6}}={a_{0}}+{a_{1}}(2x-1)+{a_{2}}{(2x-1)^{2}}+ \cdots +{a_{6}}{(2x-1)^{6}}$, find the value of $\dfrac{{a_{1}}+{a_{3}}+{a_{5}}}{{a_{0}}+{a_{2}}+{a_{4}}+{a_{6}}}$.
2. A group of 6 volunteers is to be divided into 4 teams, with 2 teams of 2 people and the other 2 teams of 1 person each, to be sent to 4 different schools for teaching. How many different allocation plans are there in total? (Answer with a number)
3. A straight line $l$ passes through the point $P(1,1)$ with an angle of inclination $\alpha = \dfrac{\pi}{6}$. The line $l$ intersects the curve $\begin{cases} x=2\cos \theta \\ y=2\sin \theta \end{cases}$ ($\theta$ is a parameter) at points $A$ and $B$. Find the value of $|PA| + |PB|$.
4. Find the distance between the centers of the two circles with polar equations $\rho = \cos \theta$ and $\rho = \sin \theta$ respectively.

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