
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  $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$ , and for  $n \ge 2$ , define  $f_n(x) = f_1(f_{n-1} (x))$ . The value of x that satisfies  $f_{1001}(x) = x - 3$  can be expressed in the form  $\frac{m}{n}$ ,
where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

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