
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
For each  $k$  , find the least  $n$  in terms of  $k$  st the following holds:
There exists  $n$  real numbers  $a_1 , a_2 ,\cdot \cdot \cdot , a_n$  st for each  $i$  :  $$ 0 < a_{i+1} - a_{i} < a_i - a_{i-1} $$   And , there exists  $k$  pairs  $(i,j)$  st  $a_i - a_j = 1$ .

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