
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
problem 1 :A sequence is defined by $ x_1 = 1, x_2 = 4$  and  $ x_{n+2} = 4x_{n+1} -x_n$  for  $n \geq 1$ . Find all natural numbers  $m$  such that the number  $3x_n^2 + m$  is a perfect square for all natural numbers  $n$ 

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