
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 a sequence \(A\): \(a_1\), \(a_2\), \(…\), \(a_n\) (\(n \geq 3\)) where \(a_i \in \mathbb{N}^* (1 \leq i \leq n)\) and for all \(2 \leq k \leq n-1\), the condition \(a_{k+1}+a_{k-1} > 2a_k\) always holds. Such a sequence is called a "U-sequence".
1. If the sequence 1, \(x\), \(y\), 7 is a "U-sequence", list all possible values of \(x\) and \(y\).
2. If for a "U-sequence" \(A\): \(a_1\), \(a_2\), \(…\), \(a_n\) we have \(a_1=1\) and \(a_n=2017\), find the maximum value of \(n\).
3. Let \(n_0\) be a given even number. For all possible "U-sequences" \(A\): \(a_1\), \(a_2\), \(…\), \(a_{n_0}\), let \(M=\max\{a_1,a_2,…,a_{n_0}\}\) where \(\max\{x_1,x_2,…,x_s\}\) denotes the largest number among \(x_1, x_2, …, x_s\). Find the minimum value of \(M\).

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