
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
Determine the general term of the sequence given by the following rules:
a) \( u_{n+3} = u_{n+2} + u_{n+1} - u_{n} \) with initial conditions \( u_0 = 0 \), \( u_1 = 1 \), \( u_2 = 2 \).

b) \( v_{n+3} = v_{n+2} - v_{n+1} + v_{n} \) with initial conditions \( v_0 = 1 \), \( v_1 = 1 \), \( v_2 = 1 \).

c) \( w_{n+3} = w_{n+2} + w_{n} \) with initial conditions \( w_0 = 1 \), \( w_1 = 1 \), \( w_2 = 2 \).

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