
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  $K > 0$  be an integer. An integer  $k \in [0,K]$  is randomly chosen. A sequence of integers is defined starting on  $k$  and ending on  $0$ , where each nonzero term  $t$  is followed by  $t$  minus the largest Lucas number not exceeding  $t$ . 

The probability that  $4$ ,  $5$ , or  $6$  is in this sequence approaches  $\tfrac{a - b \sqrt c}{d}$  for arbitrarily large  $K$ , where  $a$ ,  $b$ ,  $c$ ,  $d$ , are positive integers,  $\gcd(a,b,d) = 1$ , and  $c$  is squarefree. Find  $a + b + c + d$ .

*(Lucas numbers are defined as the members of the infinite integer sequence  $2$ ,  $1$ ,  $3$ ,  $4$ ,  $7$ ,  $\ldots$  where each term is the sum of the two before it.)*

*Proposed by Evan Chang*

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