
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 a list of numbers:
\[10, 4, 7, 4, 6, 4, x\]
when the mean, median, and mode are in increasing order, they form a non-constant arithmetic progression. Determine the sum of all possible real values of $x$.
A) $3$
B) $\frac{63}{13}$
C) $17$
D) $21$

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