
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  $a_1,a_2,a_3,\dots$  be a sequence of positive real numbers such that  $a_ka_{k+2}=a_{k+1}+1$  for all positive integers  $k$ . If  $a_1$  and  $a_2$  are positive integers, find the maximum possible value of  $a_{2014}$ .

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