
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 that the universal set is the set of real numbers \( \mathbb{R} \), let the set \( A \) be defined as \( A = \{x \mid y = \sqrt{x-1} + \sqrt{3-x}\} \), and the set \( B \) be defined as \( B = \{x \mid \log_{2}x > 1\} \).
(Ⅰ) Determine the sets \( A \cap B \) and \( (\complement_{\mathbb{R}} B) \cup A \);
(Ⅱ) Suppose the set \( C \) is defined as \( C = \{x \mid 1 < x < a\} \). If \( C \subseteq A \), find the range of the real number \( a \).

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