
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 the parabola $y=x^{2}+6x-1$.  
$(1)$ Please find the coordinates of the vertex and the axis of symmetry of the parabola using the completing the square method or the formula method.  
$(2)$ Write down the coordinates of the intersection points of the parabola with the $y$-axis.

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