
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 triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\left(1+\sin B+\cos B\right)({\cos\frac{B}{2}-\sin\frac{B}{2}})=\frac{7}{{12}}\sqrt{2+2\cos B}$. Find:

$(Ⅰ)$ $\cos B$;

$(Ⅱ)$ If $\frac{c}{a}=\frac{2}{3}$, $D$ is a point on side $AC$ such that $BD=AC$, find the value of $\frac{{AD}}{{DC}}$.

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