
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
Find the largest interval  $ M \subseteq \mathbb{R^ \plus{} }$ , such that for all  $ a$ ,  $ b$ ,  $ c$ ,  $ d \in M$  the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}\]
holds. Does the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}\]
hold too for all  $ a$ ,  $ b$ ,  $ c$ ,  $ d \in M$ ?
( $ \mathbb{R^ \plus{} }$  denotes the set of positive reals.)

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