
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
A finite set  $\mathcal{S}$  of distinct real numbers has the following properties: the mean of  $\mathcal{S}\cup\{1\}$  is 13 less than the mean of  $\mathcal{S}$ , and the mean of  $\mathcal{S}\cup\{2001\}$  is 27 more than the mean of  $\mathcal{S}.$   Find the mean of  $\mathcal{S}.$ 

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