
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 line with parametric equations \begin{cases} x=1+t \\ y=a-t \end{cases}, where $t$ is the parameter, and the circle with parametric equations \begin{cases} x=2+2\cos\alpha \\ y=2+2\sin\alpha \end{cases}, where $\alpha$ is the parameter, determine the value of $a$ such that the length of the chord intercepted by the line on the circle is $2\sqrt{2}$.

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