
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 \( n \geq 3 \) be an integer. A frog is to jump along the real axis, starting at the point 0 and making \( n \) jumps: one of length 1, one of length 2, \ldots, one of length \( n \). It may perform these \( n \) jumps in any order. If at some point the frog is sitting on a number \( a \leq 0 \), its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number \( a > 0 \), its next jump must be to the left (towards the negative numbers). Find the largest positive integer \( k \) for which the frog can perform its jumps in such an order that it never lands on any of the numbers \( 1, 2, \ldots, k \).

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