
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
Given a natural number  $n$ , if the tuple  $(x_1,x_2,\ldots,x_k)$  satisfies  $$ 2\mid x_1,x_2,\ldots,x_k $$    $$ x_1+x_2+\ldots+x_k=n $$  then we say that it's an *even partition*. We define *odd partition* in a similar way. Determine all  $n$  such that the number of even partitions is equal to the number of odd partitions.

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