
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 parallelogram $ABCD$, point $E$ is the reflection of point $B$ with respect to point $D$, and $\overrightarrow{AF}=3\overrightarrow{FC}$.  
$(1)$ Express $\overrightarrow{AE}$ and $\overrightarrow{AF}$ in terms of $\overrightarrow{AB}$ and $\overrightarrow{AD}$.  
$(2)$ If $G$ is a point on segment $EF$ such that $\overrightarrow{AG}=x\overrightarrow{AB}+y\overrightarrow{AD}$, find $5x+7y$.

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