
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 $f\left( x \right)=\left| 2x-1 \right|+\left| 2x+3 \right|.$

(1) Solve the inequality $f\left( x \right)\geqslant 6$;

(2) Let $f\left( x \right)$'s minimum value be $m$, and the positive real numbers $a,b$ satisfy $2ab+a+2b=m,$ find the minimum value of $a+2b$.

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