
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 the function $f(x)=4-|x|-|x-3|$.

$(1)$ Find the solution set for the inequality $f\left(x+ \frac{3}{2}\right) \geqslant 0$.

$(2)$ If $p$, $q$, $r$ are positive real numbers, and $\frac{1}{3p} + \frac{1}{2q} + \frac{1}{r} = 4$, find the minimum value of $3p+2q+r$.

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