
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 distribution table of the random variable $\xi$ as shown below, where $D(\xi)$ represents the variance of $\xi$, find $D(2\xi +1)$. 

| $\xi$ | $0$   | $1$       | $2$   |
|-------|-------|-----------|-------|
| $p$   | $a$   | $1-2a$    | $\frac{1}{4}$ |

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