
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 *complex set*, along with its *complexity*, is defined recursively as the following:


- The set  $\mathbb{C}$  of complex numbers is a complex set with complexity  $1$ .
- Given two complex sets  $C_1, C_2$  with complexity  $c_1, c_2$  respectively, the set of all functions  $f:C_1\rightarrow C_2$  is a complex set denoted  $[C_1, C_2]$  with complexity  $c_1 + c_2$ .
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A *complex expression*, along with its *evaluation* and its *complexity*, is defined recursively as the following:
[list]
- A single complex set  $C$  with complexity  $c$  is a complex expression with complexity  $c$  that evaluates to itself.
- Given two complex expressions  $E_1, E_2$  with complexity  $e_1, e_2$  that evaluate to  $C_1$  and  $C_2$  respectively, if  $C_1 = [C_2, C]$  for some complex set  $C$ , then  $(E_1, E_2)$  is a complex expression with complexity  $e_1+e_2$  that evaluates to  $C$ .

For a positive integer  $n$ , let  $a_n$  be the number of complex expressions with complexity  $n$  that evaluate to  $\mathbb{C}$ . Let  $x$  be a positive real number. Suppose that \[a_1+a_2x+a_3x^2+\dots = \dfrac{7}{4}.\] Then  $x=\frac{k\sqrt{m}}{n}$ , where  $k$ , $m$ , and  $n$  are positive integers such that  $m$  is not divisible by the square of any integer greater than  $1$ , and  $k$  and  $n$  are relatively prime. Compute  $100k+10m+n$ .

*Proposed by Luke Robitaille and Yannick Yao*

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