
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 polar coordinate system, the polar equations of the curves $C\_1$ and $C\_2$ are given by $ρ = -2\cos θ$ and $ρ \cos \left(\begin{matrix} \begin{matrix}θ + \dfrac{π}{3} \end{matrix}\end{matrix}\right)=1$, respectively.

1. Find the number of common points between the curves $C\_1$ and $C\_2$.
2. Draw a line through the pole intersecting the curve $C\_2$ at the point $Q$. Choose a point $P$ on $OQ$ such that $|OP| \cdot |OQ| = 2$, and find the locus equation of point $P$, then identify the shape of the locus.

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