
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
Given \( K \in \mathbf{N} \) and the experiment \( a > 0 \), under the conditions
\[ k_{1} + k_{2} + \cdots + k_{r} = k, \quad k_{i} \in \mathbf{N}, \quad 1 \leqslant r \leqslant k \],
find the maximum value of \( a^{k_{1}} + a^{k_{2}} + \cdots + a^{k_{r}} \).

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