
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 $f(x)=\dfrac{bx+1}{{(ax+1)}^{2}}(x\ne -\dfrac{1}{a},a > 0)$, and $f(1)=\dfrac{1}{4}$, $f(-2)=1$.

$(1)$ Find the expression for the function $f(x)$;

$(2)$ Given that the sequence $\{x_n\}$ satisfies $x_n=[1-f(1)]\cdot[1-f(2)]\cdot…\cdot[1-f(n)]$, try to find $x_1$, $x_2$, $x_3$, $x_4$;

$(3)$ Conjecture the general term of $\{x_n\}$.

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