
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
$(1)$ Given $f\left(x+1\right)=x^{2}+4x+1$, find the analytical expression for $f\left(x\right)$;<br/>$(2)$ Given $f\left(x\right)+3f\left(-x\right)=x^{2}-2x$, find the analytical expression for $f\left(x\right)$;<br/>$(3)$ Given $f\left(x\right)$ is a linear function and satisfies $3f\left(x+1\right)-f\left(x\right)=2x+9$.

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