
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
"Modulo $6$ graph paper" consists of a grid of $36$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <6$. To graph a congruence on modulo $6$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Plot the graph of the congruence $$y\equiv 4x^2 + 3x - 1 \pmod{6}$$ and find the total number of points satisfying this congruence.

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