
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
Triangles $PQR$ and $PST$ have areas $1234$ and $4321,$ respectively, with $Q=(0,0), R=(134,0), S=(400,250),$ and $T=(410,260).$ What is the sum of all possible $x$-coordinates of $P$?

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