
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
Given  $X=\{0,a,b,c\}$ , let  $M(X)=\{f|f: X\to X\}$  denote the set of all functions from  $X$  into itself. An addition table on  $X$  is given us follows:
 $+$     $0$    $a$    $b$   $c$ 
 $0$     $0$    $a$    $b$   $c$ 
 $a$     $a$    $0$    $c$   $b$ 
 $b$     $b$    $c$    $0$   $a$ 
 $c$     $c$    $b$    $a$   $0$ 
a)If  $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$ , find  $|S|$ .
b)If  $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$ , find  $|I|$ .

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