
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 $C$: $y^{2}=4x$.
(1) A point $P$ on the parabola $C$ has a vertical line $PQ$ drawn to the $x$-axis, with foot $Q$. Determine the trajectory $D$ of the midpoint $R$ of $PQ$.
(2) A line $l$ with a slope angle of $45^{\circ}$ passes through the focus of parabola $C$, intersecting trajectory $D$ at points $A$ and $B$. Calculate the value of $|AB|$.

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