
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
Determine the smallest integer $N$ such that it is possible to find 125 distinct integers $a_{1}, a_{2}, \ldots, a_{125}$ in a sequence such that:

(1) each of these integers is strictly positive and less than or equal to $N$;

(2) each of the 123 integers $a_{2}, \ldots, a_{124}$ is strictly greater than the arithmetic mean of the integer written to its left and the integer written to its right.

Note: The arithmetic mean of $a$ and $b$ is the number $\frac{a+b}{2}$.

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