
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  $n$  be an positive integer. Find the smallest integer  $k$  with the following property; Given any real numbers  $a_1 , \cdots , a_d $  such that  $a_1 + a_2 + \cdots + a_d = n$  and  $0 \le a_i \le 1$  for  $i=1,2,\cdots ,d$ , it is possible to partition these numbers into  $k$  groups (some of which may be empty) such that the sum of the numbers in each group is at most  $1$ .

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