
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 $f(x)=|2x+1|-|x-2|$.  
$(1)$ Find the solution set $M$ for $f(x)\leqslant 2$;  
$(2)$ Let the maximum element of set $M$ be $m$. If $a$, $b$, and $c$ are all positive real numbers, and $\frac{1}{a}+\frac{1}{{2b}}+\frac{1}{{3c}}=m$, find the minimum value of $a+2b+3c$.

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