
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
Rational numbers $m$, $n$, and $p$ that are not equal represent points $M$, $N$, and $P$ on the number line, respectively. If $MP=PN=r$, then $m$ and $n$ are said to be symmetric about $p$ with a symmetric radius of $r$. For example, rational numbers $3$ and $5$ are symmetric about $4$ with a symmetric radius of $1$.

$(1)$ If rational numbers $3$ and $x$ are symmetric about $1$, then $x=$______; the symmetric radius $r=$______;

$(2)$ If rational numbers $a$ and $b$ are symmetric about $2$, and $|a|=2|b|$, find the symmetric radius $r$.

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