
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 $\{a_{n}\}$ is a geometric sequence, $\{b_{n}\}$ is an arithmetic sequence, and $a_{1}=2$, $b_{1}=1$, $a_{3}b_{4}=2a_{4}$, $a_{2}=b_{1}+b_{3}$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Find $\sum_{k=1}^{{2^n}-1}{{b_k}}$;  
$(3)$ Let the general formula for the sequence $\{d_{n}\}$ be: ${d_n}=\left\{{\begin{array}{l}{-\frac{{{b_n}{{({{a_n}})}^2}}}{2}, \text{if } n \text{ is odd}}\\{\frac{{{b_n}{{({{a_n}})}^2}}}{4}, \text{if } n \text{ is even}\end{array}}\right.$. Find $\sum_{i=1}^{2n}{{d_i}}$.

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