
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
Find the least positive integer \( N > 1 \) satisfying the following two properties:
- There exists a positive integer \( a \) such that \( N = a(2a - 1) \).
- The sum \( 1 + 2 + \cdots + (N - 1) \) is divisible by \( k \) for every integer \( 1 \leq k \leq 10 \).

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