
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
For any positive integer $n$, let $f_{1}(n)$ denote the square of the sum of the digits of $n$ plus $r+1$, where $r$ is the integer satisfying $n = 3q + r$ with $0 \leqslant r < 3$. For $k \geqslant 2$, let $f_{k}(n) = f_{1}(f_{k-1}(n))$. Find $f_{1990}(2345)$.

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