
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
Solve the following equations:<br/>$(1)$ Solve the quadratic equation using the completing the square method: $x^{2}-2x-2=0$;<br/>$(2)$ Solve the equation using the factoring method: $\left(x+4\right)^{2}=5\left(x+4\right)$;<br/>$(3)$ Solve the equation using the formula method: $3x^{2}+6x-5=0$;<br/>$(4)$ Solve the equation using an appropriate method: $4x^{2}+2x=1$.

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