
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 $m$ graph paper" contains $m^2$ points representing all pairs of integer residues $(x,y)$ where $0\le x<m$. When graphing a congruence on "modulo $m$ graph paper," we mark every point $(x,y)$ that satisfies the congruence. Consider demonstrations on modulo $25$ graph paper for the graphs of $$y\equiv 10x+3\pmod{25}$$ and $$y\equiv x^2 + 15x + 20 \pmod{25}.$$ Identify the points the graphs have in common, and sum the $x$-coordinates of these points.

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