
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
There exists a unique strictly increasing arithmetic sequence  $\{a_i\}_{i=1}^{100}$  of positive integers such that \[a_1+a_4+a_9+\cdots+a_{100}=\text{1000},\] where the summation runs over all terms of the form  $a_{i^2}$  for  $1\leq i\leq 10$ .  Find  $a_{50}$ .

*Proposed by David Altizio and Tony Kim*

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