
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 plane rectangular coordinate system ${xOy}$, the parametric equations of the line ${l}$ are ${\left\{ \begin{matrix}x = - \dfrac{1}{2} + \dfrac{\sqrt{2}}{2}t \\y = 1 + \dfrac{\sqrt{2}}{2}t \end{matrix} \right.}$ (${t}$ is the parameter), and the equation of the circle ${C}$ is ${(x-2)^{2}+ (y-1)^{2}= 5}$. Establish a polar coordinate system with the origin ${O}$ as the pole and the positive x-axis as the polar axis. (${1}$) Find the polar coordinate equations of the line ${l}$ and the circle ${C}$; (${2}$) If the line ${l}$ intersects the circle ${C}$ at points ${A}$ and ${B}$, find the value of ${\cos \angle AOB}$.

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