
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
Define the function \( f_1(x) = \frac{1}{2} - \frac{4}{4x+2} \), and for \( n \geq 2 \), define \( f_n(x) = f_1(f_{n-1}(x)) \). Find the value of \( x \) that satisfies \( f_{1001}(x) = x-2 \).

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