
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
For \( k, n \in \mathbf{Z}_{+} \), consider the finite sequence \(\{a_k\}\) of length \( n \) with \( a_k \leq m \) and \( a_k, m \in \mathbf{Z}_{+} \). Determine the maximum value of \( n \) such that the sequence \(\{a_k\}\) satisfies the following conditions:
(1) For any term \( a_k \), if \( a_{k-1} \) and \( a_{k+1} \) exist, then \( a_{k-1} \neq a_{k+1} \);
(2) There do not exist positive integers \( i_1 < i_2 < i_3 < i_4 \) such that \( a_{i_1} = a_{i_3} \neq a_{i_2} = a_{i_4} \).

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