
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
Let  $ \{m_1,m_2,\dots\}$  be a (finite or infinite) set of positive integers. Consider the system of congruences 

(1)    $ x\equiv 2m_i^2 \pmod{2m_i\minus{}1}$  ( $ i\equal{}1,2,...$  ).

Give a necessary and sufficient condition for the system (1) to be solvable.

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