
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 given positive integer, and let \( a_{1}, a_{2}, \cdots, a_{n} \) be a sequence of real numbers such that for each \( m \leq n \), the inequality \(\left|\sum_{k=1}^{m} \frac{a_{k}}{k}\right| \leq 1\) holds. Determine the maximum value of \(\left|\sum_{k=1}^{n} a_{k}\right|\).

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