
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  $N=10^6$ . For which integer  $a$  with  $0 \leq a \leq N-1$  is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?

*Proposed by Lewis Chen*

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