
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" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0 \leq x < m$. The graphs of $$y \equiv 3x + 1 \pmod{11}$$ and $$y \equiv 7x + 5 \pmod{11}$$ on modulo $11$ graph paper have some points in common. What is the sum of the $x$-coordinates of those points?

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