
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
The quadratic function $y=ax^2+bx+c$ passes through points A and B (A is to the left of B), and they are distributed on both sides of the y-axis. The lengths of OA and OB are the two roots of the equation $x^2-5x+4=0$, and $OA > OB$. The function intersects the y-axis at point C $(0, 4)$.  
(1) Find the value of $4a-2b+c$;  
(2) Connect AC and BC. Let P be a moving point on segment AB, and $AP=m$. Draw $PM \parallel AC$, intersecting BC at M. What value of $m$ makes the area of $\triangle PCM$ the largest, and what is this maximum value?  
(3) The area of the circumscribed circle of $\triangle ABC$ is $\boxed{\frac{17}{2}\pi}$.

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