
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
A positive integer  $K$  is given. Define the sequence  $(a_n)$  by  $a_1 = 1$  and  $a_n$  is the  $n$ -th positive integer greater than  $a_{n-1}$  which is congruent to  $n$  modulo  $K$ .**(a)** Find an explicit formula for  $a_n$ .**(b)** What is the result if  $K = 2$ ?

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