
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
Points $A$ and $B$ correspond to numbers $x$ and $y$ on the number line, respectively, where $x$ and $y$ satisfy $|x-3|+|y+5|=0$.  
$(1)$ Find the values of $x$ and $y$;  
$(2)$ There is a point $M$ on the number line such that $|AM|+|BM|=\frac{7}{4}|AB|$. Find the number corresponding to point $M$;  
$(3)$ Point $D$ is the midpoint of $AB$, $O$ is the origin. There is a moving point $P$ on the number line. Write down the minimum value of $|PA|+|PB|$ as ______; the minimum value of $|PD|-|PO|$ as ______; the range of values of number $a$ corresponding to point $P$ when $|PA|+|PB|+|PD|-|PO|$ is minimized is ______.

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