
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
If the curve $C_1: \theta = \frac{\pi}{6} (\rho \in \mathbb{R})$ intersects with the curve $C_2: \begin{cases} x = a + \sqrt{2}\cos\theta \\ y = \sqrt{2}\sin\theta \end{cases}$ ($\theta$ is the parameter, $a$ is a constant, $a > 0$) at two points A and B, and the distance $|AB| = 2$, find the value of the real number $a$.

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