
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
On "Modulo $13$ graph paper" which has $13^2$ points, graph the congruences:
$$y \equiv 3x + 4 \pmod{13}$$
and
$$y \equiv x^2 + x + 1 \pmod{13}.$$
Identify all points $(x, y)$ that satisfy both congruences and find the sum of the $x$-coordinates of these points.

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