
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
For a positive integer \( n \), find the smallest integer \( k \) such that for any given real numbers \( a_{1}, a_{2}, \cdots, a_{d} \) satisfying
$$
\begin{array}{l}
a_{1} + a_{2} + \cdots + a_{d} = n, \\
0 \leqslant a_{i} \leqslant 1 \quad (i = 1, 2, \cdots, d),
\end{array}
$$
these real numbers can be divided into \( k \) groups (where some groups may be empty), such that the sum of each group is at most \( 1 \).

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