when you are asked to design a dynamic programming algorithm, you should answer the following questions.

$($i$)$ What are the sequence decision steps?

$($ii$)$ What are the alternatives at each decision step?

$($iii$)$ Describe the subproblems.

$($iv$)$ Write the recurrence relation. Explain why it is correct.

$($v$)$ Describe the table that is necessary for the dynamic programming algorithm.

$($vi$)$ Describe how to fill the table.

$($vii$)$ Write down the pseudo$-$code of the dynamic programming algorithm to fill the table.

$($viii$)$ Analyze the running time efficiency of your algorithm.

$($ix$)$ Analyze the space efficiency.

$($x$)$ Write down the pseudo$-$code of an algorithm to find the actual solution for the problem Consider a sequence of positive integers $a_1,a_2,$dots,$a_n$ where

$a_1=1$ and

$a_{n}ma{x}_{1}in-1(ai+1)$

Problem: To find the maximum number of such sequences with a length of $L.$

Hint: $1a_{L}L,$ where $a_L$ is the last element and $L$ is the length of the sequence.

Example:

Input: $L=3$

There are $5$ possible sequences: $1,1,1$

$1,1,2$

$1,2,1$

$1,2,2$

$1,2,3$

Design a DP algorithm to solve the problem.