
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
Let the function $f(x)=|x-a|$ where $a\in \mathbb{R}$.

(I) When $a=3$, solve the inequality $f(x) \geqslant 4-|x+1|$;

(II) If the solution set of the inequality $f(x) \leqslant 1$ is $[1,3]$, and $\dfrac{1}{m} + \dfrac{1}{2n} = a$ where $m > 0, n > 0$, find the minimum value of $m+2n$.

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