
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
Let  $S$  be the set  $\{1, 2, ..., 10\}$ . Let  $A$  be a subset of  $S$ .
We arrange the elements of  $A$  in increasing order, that is,  $A = \{a_1, a_2, ...., a_k\}$  with  $a_1 < a_2 < ... < a_k$ . 
Define *WSUM* for this subset as  $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$  where the first term contains the odd numbered terms and the second the even numbered terms. 
(For example, if  $A = \{2, 5, 7, 8\}$ , *WSUM* is  $3(2 + 7) + 2(5 + 8)$ .) 
Find the sum of *WSUMs* over all the subsets of S. 
(Assume that WSUM for the null set is  $0$ .)

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