
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 $f\left(x\right)=|x+1|-2|x-2|$.  
$(1)$ Find the solution set $M$ of the inequality $f\left(x\right)\geqslant -1$;  
$(2)$ Let the largest integer in set $M$ from (1) be $t$. If positive numbers $a$, $b$, and $c$ satisfy $4a+b+c=t$, find the minimum value of $\frac{1}{2a+b}+\frac{1}{2a+c}$.

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