
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
The *equatorial algebra* is defined as the real numbers equipped with the three binary operations  $\natural$ ,  $\sharp$ ,  $\flat$  such that for all  $x, y\in \mathbb{R}$ , we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\]
An *equatorial expression* over three real variables  $x$ ,  $y$ ,  $z$ , along with the *complexity* of such expression, is defined recursively by the following:


- $x$ ,  $y$ , and  $z$  are equatorial expressions of complexity 0;
- when  $P$  and  $Q$  are equatorial expressions with complexity  $p$  and  $q$  respectively, all of  $P\mathbin\natural Q$ ,  $P\mathbin\sharp Q$ ,  $P\mathbin\flat Q$  are equatorial expressions with complexity  $1+p+q$ .

Compute the number of distinct functions  $f: \mathbb{R}^3\rightarrow \mathbb{R}$  that can be expressed as equatorial expressions of complexity at most 3.

*Proposed by Yannick Yao*

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