
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
An  $n$  by  $n$  grid, where every square contains a number, is called an  $n$ -code if the numbers in every row and column form an arithmetic progression.  If it is sufficient to know the numbers in certain squares of an  $n$ -code to obtain the numbers in the entire grid, call these squares a key.

**a.)**Find the smallest  $s \in \mathbb{N}$   such that any  $s$  squares in an  $n-$ code  $(n \geq 4)$  form a key.

**b.)** Find the smallest  $t \in \mathbb{N}$  such that any  $t$  squares along the diagonals of an  $n$ -code  $(n \geq 4)$  form a key.

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