
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 variant of the Fibonacci sequence where it starts with \( F_1 = 2 \) and \( F_2 = 2 \). Each following term is still the sum of the previous two terms. Calculate the remainder when the \( 50^{\mathrm{th}} \) term of this sequence is divided by 5. Additionally, compute the remainder of the \( 50^{\mathrm{th}} \) term when divided by 7, and determine if the two remainders are equal.

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