
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 $X$ and $Y$ are independent variables, the distribution of random variable $X$ is $(\begin{array}{l}{0}&{1}\\{\frac{1}{2}}&{a}\end{array})$, and the distribution of random variable $Y$ is $(\begin{array}{l}{1}&{2}\\{b}&{\frac{1}{3}}\end{array})$, find $D\left[X+2Y\right]=\_\_\_\_\_\_$.

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