# Numerical Palindromes

Let $$N$$ be the $$1500$$-digit string of decimal digits linked here.

Consider all substrings of $$N$$ with length strictly greater than $$1.$$ Let $$P$$ be the number of these substrings that are palindromes (read the same forwards and backwards), and let $$L$$ be the largest palindromic substring (converted to an integer). Compute the remainder when $$P + L$$ is divided by $$1000.$$

EDIT: The number is:

  845339512654353013973726529001291079167398839841139949030851696390946980101698124639149591440119733604468880607555560607206064491347909369911515301019367711551352591990261320383329288164577799740443242050242319267457621744807851983932903020081029321152325175250685866007841518940360280269649423980138752509028919270696197406591840726262868909999601238346887569491166411930108771076009573412594100219771074212035289275491573364890832367256854827355325731308226389028814609823679403212837077171147035094962722853937529780881806452156815760851836710785281112178566572142070924792670573967021943625926183861042230278394450084283686782710687142035589512463860007382666540574865786548090984873591052303316826223089810233823833877818493413413377572125316440837829248059997597530413074225936022537356928726861032836482495816729317138037356704312711814039586229918016418907376961773423879993723686730471541871754763457516922397978666764180907493617108196535775671577122375548642715864362660363698285254692252450789495296857844775409285832067120397741225920809730915094640238325092213217491635025536097960140990339793121105117908393671593840606430927286507282838120710540761234699189422511628632095218354056738982111107254470000373451953905544838296347383706816543780729267523585541214385956448406268928721934984154648198940685892036868554606054558448653750977153318569536525693630516086917674823412432181461884413411201293004004992015882876179775802939973661220145456423636324708339384632070192417332964444202

×