
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 $C: y^{2}=4x$, a line passing through point $A(-1,0)$ intersects the parabola $C$ at points $P(x_{1},y_{1})$ and $Q(x_{2},y_{2})$. It is given that $\overrightarrow{AP}=\lambda \overrightarrow{AQ}$.  
(Ⅰ) Find the values of $x_{1}$ and $x_{2}$ in terms of $\lambda$;  
(Ⅱ) If $\lambda \in \left[ \frac{1}{3}, \frac{1}{2} \right]$, find the equation of line $PQ$ when $|PQ|$ is maximized.

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