
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
You are given  $n$  not necessarily distinct real numbers  $a_1, a_2, \ldots, a_n$ . Let's consider all  $2^n-1$  ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to  $1$ ?

For example, if  $a = [-1, 2, 2]$ , then we got  $3$  once,  $4$  once,  $2$  twice,  $-1$  once,  $1$  twice, so the total number of ones here is  $2$ .

*(Proposed by Anton Trygub)*

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