
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 the parabola $y^{2}=2px (p > 0)$ with focus $F$, a line $l$ passes through the fixed point $A(1,0)$ and intersects the parabola at points $P$ and $Q$.
(1) If a circle passing through the origin $O$ has chord $PQ$ as its diameter, find the value of $p$.
(2) Under the conditions of (1), if $\overrightarrow{FP} + \overrightarrow{FQ} = \overrightarrow{FR}$, find the trajectory equation of the moving point $R$.

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