
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
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$ respectively. It is known that $\frac{{\cos A}}{{1+\sin A}}=\frac{{\sin 2B}}{{1+\cos 2B}}$.  
$(1)$ If $C=\frac{{2\pi}}{3}$, find $B$;  
$(2)$ Find the minimum value of $\frac{{{a^2}+{b^2}}}{{{c^2}}}$.

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