
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
The sum of the even integers from 2 to \( 2k \) inclusive equals \( t \) for some positive integer \( k \). That is,

\[ 2 + 4 + 6 + \cdots + (2k - 2) + 2k = t \]

What is the value of \( k \)?

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