
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
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If  $a$  is a ridiculous number, then  $\sqrt{a}$  and  $1+\sqrt{a}$  are also ridiculous numbers. 
[/list]
A closed interval  $I$  is ``boring'' if


- $I$  contains no ridiculous numbers, and
- There exists an interval  $[b,c]$  containing  $I$  for which  $b$  and  $c$  are both ridiculous numbers.

The smallest non-negative  $l$  such that there does not exist a boring interval with length  $l$  can be represented in the form  $\dfrac{a + b\sqrt{c}}{d}$  where  $a, b, c, d$  are integers,  $\gcd(a, b, d) = 1$ , and no integer square greater than 1 divides  $c$ . What is  $a + b + c + d$ ?

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