
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 a positive integer. Find the smallest integer \( k \) with the following property: Given any real numbers \( a_{1}, \ldots, a_{d} \) such that \( a_{1} + a_{2}+ \cdots+ a_{d} = n \) and \( 0 \leq a_{i} \leq 1 \) for \( i=1,2,\ldots,d \), 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}
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