
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
Consider the parabola  $C: y^{2}=4 x$  and the straight line  $L: y=x+2$ . Let  $P$  be a variable point on  $L$ . Draw the two tangents from  $P$  to  $C$  and let  $Q_{1}$  and  $Q_{2}$  denote the two points of contact on  $C$ . Let  $Q$  be the mid-point of the line segment joining  $Q_{1}$  and  $Q_{2}$ . Find the locus of  $Q$  as  $P$  moves along  $L$ .

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