
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 the Cartesian coordinate system, there is a line $m$: $ax-3y+2=0$.
$(1)$ If the intercept of line $m$ on the $x$-axis is $-2$, find the value of the real number $a$ and write the intercept form of the equation of line $m$;
$(2)$ If a line $n$ passes through point $M(3,1)$ and is parallel to line $m$, and the equation of line $n$ is $4x-6y+b=0$, find the values of the real numbers $a$ and $b$, and calculate the distance between the two parallel lines $m$ and $n$.

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