
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
Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in $\triangle ABC$, and $D$ is the midpoint of side $AC$, with $a=BD=4\sqrt{7}$ and $c=12$. <br/>$(1)$ Find the value of $b$; <br/>$(2)$ If the angle bisector of $\angle BAC$ intersects $BC$ at point $E$, find the length of segment $AE$.

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