
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 that $f(x) = |x - a|$ where $a > 0$.

(I) Find the value of the real number $a$ if the minimum value of the function $F(x) = f(x) + f(2x)$ is 3.

(II) If $a = 2$, the maximum value of the function $g(x) = f(x) - f(-x)$ is $k$, and $2m + 3n = k$ where $m > 0$ and $n > 0$. Find the minimum value of $\frac{1}{m} + \frac{2}{3n}$.

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