
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
Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ adding 300 more elements to the union, how many elements are in set $A$?

[asy]
label("$A$", (2,67));
label("$B$", (80,67));
label("$C$", (41,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
draw(Circle((44, 27), 22));
label("1200", (44, 45));
label("300", (44, 27));
[/asy]

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