
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 parabolas $C_{1}$: $y=x^{2}+2x$ and $C_{2}$: $y=-x^{2}+a$, if line $l$ is a common tangent of both $C_{1}$ and $C_{2}$, find the value of $a$ such that $C_{1}$ and $C_{2}$ have exactly one common tangent. Write out the equation of the common tangent.

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