
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
A finite set  $\{a_1, a_2, ... a_k\}$  of positive integers with  $a_1 < a_2 < a_3 < ... < a_k$  is named  *alternating* if  $i+a$  for  $i = 1, 2, 3, ..., k$  is even. The empty set is also considered to be alternating. The number of alternating subsets of  $\{1, 2, 3,..., n\}$  is denoted by  $A(n)$ .
Develop a method to determine  $A(n)$  for every  $n \in N$  and calculate hence  $A(33)$ .

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