
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 the sequence: $$( \frac{1}{1}), ( \frac{1}{2}, \frac{2}{1}), ( \frac{1}{3}, \frac{2}{2}, \frac{3}{1}), ( \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1}), \ldots, ( \frac{1}{n}, \frac{2}{n-1}, \frac{3}{n-2}, \ldots, \frac{n-1}{2}, \frac{n}{1})$$, denote the elements of the sequence as: $a_1, a_2, a_3, a_4, a_5, \ldots$ then $a_{2009} = \boxed{?}$.

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