
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
Through the point \( P(11,0) \), draw a line with an angle of inclination \( \frac{\pi}{4} \) intersecting the parabola \( y^{2}=4x \) at points \( R \) and \( Q \). Then, draw a line parallel to \( RQ \) intersecting the arc of the parabola \( \overparen{ROQ} \) at points \( M \) and \( N \). (As shown in the diagram), find the maximum area of the triangle \( \triangle PMN \).

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